Method and Apparatus for Embossing a Deformable Body

ABSTRACT

A deformable body is embossed using a stamp as a function of a pressure distribution. The pressure distribution is obtained by running a deformation model determined based on convolving a contact pressure distribution with a mechanical response of a surface topography of the deformable body and convolving the pressure distribution with a point load response of the stamp.

RELATED APPLICATIONS

This application is a continuation-in-part of U.S. application Ser. No.12/943,889, filed Nov. 10, 2010, which claims the benefit of U.S.Provisional Application No. 61/259,860, filed on Nov. 10, 2009.

The entire teachings of the above applications are incorporated hereinby reference.

BACKGROUND

The hot embossing of thermoplastic polymers has attracted attention as apromising microfabrication process. Hot embossing has certain advantagesover other polymer microfabrication processes. The micro-casting ofcurable liquid resins, which is a process that is used with elastomerssuch as polydimethylsiloxane (PDMS), is widely known as soft lithography(1) and is ideal for prototyping small numbers of devices.Unfortunately, considerable manual skill is required to handle thehighly flexible components produced.

Available techniques for automating soft lithography have so far provedlargely elusive. One example of such methods is injection molding.Injection molding may be used to form microscopic features (2) and caneasily be automated, but tooling and equipment costs associated withthis method are relatively high.

Finite-element numerical modeling of thermoplastic embossing has alsoreceived attention in the art. For example, patterning ofsub-micrometer-thickness polymeric layers, as encountered innano-imprint lithography, has been considered (8-14). The embossedmaterial has variously been described using models such as Newtonianliquid (8, 9, 14), shear-thinning liquid (8, 15), linear-elastic (11),Mooney-Rivlin rubber-elastic model (10, 16), and linear (12, 13) ornon-linear (17, 18) visco-elastic models. Otherthermomechanically-coupled finite-deformation material models have alsobeen developed (19, 20) and applied to simulate the micro-embossing ofbulk polymeric substrates (20). However, finite-element approaches,although capable of capturing many of the physical phenomena observed,are currently too computationally costly to extend to the feature-richpatterns of complete devices.

For the simulation of nanoimprint lithography, Zaitsev, et al. (22) haveproposed a simplified “coarse-grain” approach in which the imprintedpolymeric layer is modeled as a Newtonian fluid and the pattern of thestamp is represented by a matrix of cells, where each is assumed tocontain features of a single size and packing density (21-25).

Efficient numerical simulations of the deformations of elastic (26, 27)and elastic-plastic (28-30) bodies, which may be rough and/ormulti-layered (26, 29, 30), have also been considered in tribology.These simulations, in the elastic-plastic cases, rely on a descriptionof the deformation of the material's surface in response to a point-loadtogether with a criterion for yielding of the material. The overalltopography of the material's surface is calculated by spatiallyconvolving an iteratively-found contact pressure distribution with thepoint-load response. Sub-surface stresses can similarly be estimated byconvolving contact pressures with appropriate kernel functions (27). Theconvolution itself may be effected using fast Fourier transforms (26,28, 29) or other summation methods (31, 32). The solution for thecontact-pressure distribution may successfully be obtained usingiterative conjugate-gradient methods combined with kinematic constraintson the surface deformation (26, 28, 32) or by using methods that seek aminimum of elastic potential energy in the layer (29).

The validity of these contact mechanics-based approaches is limited tocases where surface curvatures remain small and all deflections are asmall proportion of any layer's thickness. These linear methods havenevertheless proved to be of great value because of the fast computationthat is possible. Lei, et al. suggest using such an approach torepresent the micro-embossing of thick, rubbery polymeric layers (33).They develop an approximate analytical expression for the shape of thedeformed surface of such a layer when embossed with a simple trench, andshow rough agreement between that expression and the measured topographyof polymethylmethacrylate layers embossed under a small set ofprocessing conditions.

SUMMARY

Certain aspects of the present invention relate to a method andcorresponding apparatus for embossing a deformable body with a stampthat runs a deformation model determined based on convolving a contactpressure distribution with a mechanical response of a surface topographyof the deformable body and convolving the pressure distribution with apoint load response of the stamp and embosses the deformable body usingthe stamp as a function of the deformation model.

Yet another aspect of the present invention relates to a method andcorresponding apparatus for embossing a deformable body with a stampthat runs a deformation model determined based on convolving a contactpressure distribution and a point load response. The point load responsemay be an anisotropic function. The deformable body may be embossedusing the stamp as a function of the deformation model.

The point load response may be determined based on deflections of thestamp. The deformable body may include a polymeric layer comparable inthickness to lateral dimensions of features being embossed. Thedeformable body may include a polymeric layer being of thicknesssubstantially less than lateral dimensions of features being embossed.The deformable body includes a deformable layer on a substrate. Thedeformable layer may be unitary with the substrate, be bounded with thesubstrate, or be mechanically supported on the substrate.

The deformation model may further be obtained by convolving the contactpressure distribution with a point load response of the substrate.

The contact pressure distribution may be determined based on stampindentation and substrate indentation. The contact pressure distributionmay be determined based on deformations of a layer of the deformablebody. The contact pressure distribution may be determined based on azero-mean pressure distribution needed to bring all surfaces of thelayer of the deformable body into contact with the stamp. The contactpressure distribution may be determined based on a zero-mean pressuredistribution needed for an incremental displacement of the stamp intothe deformable body. The contact pressure distribution may be determinedbased on a spatial variation of thickness of the layer of the deformablebody or based on the point load response and an average pressure appliedto the stamp.

The mechanical response of the surface topography of the deformable bodymay be anisotropic.

The point load response of the stamp may be determined based on bendingand/or indentation of the stamp. The bending of the stamp may bedetermined using at least one of the thickness and the elasticity of thestamp.

The deformable body may include a polymeric layer thicker thandimensions of features being embossed.

The displacement of material in the deformable body embossed with thestamp may be determined as a function of the point-load-time response,one or more additional properties of the deformable body, and thecontact pressure distribution. The one or more additional properties ofthe deformable body may include at least one of time dependentproperties, temperature dependent properties, temperature and timedependent properties, temperature dependent elasticity, or temperaturedependent viscosity.

The anisotropic point load response may be obtained using an anisotropyparameter that represents strength of anisotropy of the anisotropiclayer of deformable body.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing will be apparent from the following more particulardescription of example embodiments of the invention, as illustrated inthe accompanying drawings in which like reference characters refer tothe same parts throughout the different views. The drawings are notnecessarily to scale, emphasis instead being placed upon illustratingembodiments of the present invention.

FIG. 1 illustrates an example of hot embossing of a deformable body.

FIG. 2 is a plot that illustrates a relationship between time,temperature, loading duration, and applied load in hot embossing ofthermoplastic polymers.

FIG. 3 a is an illustration of representation of global compression offinite-thickness substrates before loading.

FIG. 3 b is an illustration of representation of global compression offinite-thickness substrates under uniform applied pressure in the zdirection.

FIGS. 4 a, 4 b, and 4 c illustrate an anisotropic point-load and itsrelation to geometry of an embossed pattern.

FIGS. 5 a-5 f are plots that illustrate measured peak cavity penetrationheights patterns of two samples embossed using aspects of the presentinvention.

FIG. 6 is a table that includes an outline of some possible polymermodels and their corresponding compliance functions.

FIG. 7 illustrates Kernel functions used for describing stamp bending incertain aspects of the present invention.

FIG. 8 is a table that includes values required for construction of aplate bending function Kernel.

FIG. 9 is an illustration of contact pressure non-uniformity for a hardrough surface and a stamp that can either bend or be indented.

FIG. 10 is a schematic graph of the maximum surface waviness to which astamp can conform.

FIG. 11 is a plot that illustrates the ability of a two-layer stamp toconform to surface roughness.

FIG. 12 is a table that includes initial values of variables involved innanoimprint simulation.

FIGS. 13 a-13 c illustrate optical micrographs and white-lightinterferometry cross-sections of test patterns.

FIG. 14 is a plot that illustrates determination of initial PMMA layerthickness using scanning white-light interferometry results.

FIGS. 15 a-15 c illustrate results of the cavity filled simulation.

FIG. 16 illustrates a region of laterally enclosed resist in a filledcavity or cavities.

FIGS. 17 a-17 b illustrate a generalization of the simulation ofthin-layer squeezing for varying layer thicknesses and lateralconstraints (1-D case).

FIGS. 18 a-18 e includes examples of uses of the decomposed-kernelapproach to simulate several one-dimensional geometries.

FIG. 19 is a flow diagram of an example embodiment of the presentinvention for modeling deformation of a deformable body embossed with astamp.

FIG. 20 is a flow diagram of an example embodiment of the presentinvention for determining a displacement of material in an embossedsubstrate.

FIG. 21 is a flow chart of an example embodiment of the presentinvention for determining a displacement of a material in athermoplastic embossed with a stamp.

DETAILED DESCRIPTION

A description of example embodiments of the invention follows.

Certain aspects of the invention model the deflections of stamp andsubstrate that govern the amount of residual layer non-uniformity anddistinguish between a stamp's bending and its local indentation,offering insights for the selection of stamp materials and for theengineering of multi-layered stamps.

Certain aspects of the invention provide layout ‘design rules’ that mayassist device designers in minimizing residual layer thicknessvariation.

The set of techniques described herein forms a basis for a family ofprocess-design and pattern-refinement software tools. In doing so, thesemodeling approaches may be the key to much broader industrial acceptanceof hot micro-embossing and of nanoimprint lithography.

FIG. 1 illustrates an example 100 of hot embossing of a deformable bodysuch as a thermoplastic polymer 110. In hot embossing, a deformable body(e.g., polymer) 110 is heated until it softens, and a hard, reusable,patterned stamp 120 is then pressed into the polymer 110 before both thestamp 120 and the polymer 110 are cooled and then separated. The heatingof the polymer 110 may be performed using a heated platen 140 or usingany other available technique in the art. The deformable body 110 may beheated above its glass-transition temperature and an embossing load 130may be applied in order to transfer a microstructure from the stamp 120to the softened polymer 110. An elastometric gasket 150 may be employedon the flat side of the polymer 110 to ensure that the applied load isdistributed adequately and uniformly across the flat side of the polymereven if the heated platens are not perfectly parallel. The polymer 110may then be cooled to below its glass-transition temperature and theload 130 may be removed and separated from the stamp 120.

The polymer 110 may alternatively be patterned on both sides byinserting a second patterned stamp between the polymer 110 and the rigidplate. Alternatively, the rigid plate could be replaced by a secondpatterned stamp. Any such second stamp may carry the same pattern asstamp 120, or alternatively a different pattern.

Hot embossing combines moderate cost with ease of automation and mayhelp bridge the gap between the invention and the commercialization of anumber of micro and nano-fluidic devices and other micro- andnano-devices. Substrates processed by hot embossing may range in sizefrom a single chip to a continuous roll of material (3-5). Thesefeatures may make hot embossing both amenable to prototyping andpotentially more cost-efficient than techniques such as injectionmolding for very high-volume manufacturing.

Hot embossing for the fabrication of microelectromechanical systems(MEMS) or microfluidics is usually performed on homogeneous polymericsheets that are much thicker than the characteristic feature sizes ofthe patterns being embossed. These embossed layers constitute the bodyof the device being manufactured.

FIG. 2 is a plot that illustrates a relationship between time 205,temperature 210, loading duration 230, and applied load 240 in hotembossing of thermoplastic polymers, according to an example embodimentof the present invention. A pattern to be hot embossed may containthousands of features ranging from less than a micron to severalmillimeters in diameter. An embossing temperature 210, load 220, andloading duration 230 must be selected such that every cavity on thestamp is filled with polymer as required. Other constraints on themaximum load 240 and temperature 210 may also be applied. For example,factors such as design of the embossing apparatus or an embosser, desireto restrict differential thermal contraction of stamp and substrate(6-7), or possibility of substrate degradation at very high temperaturesmay be included as limiting factors on the maximum load and temperature.

The loading duration 230 may be constrained by a desire to maximizethroughput. One example embodiment of the present invention may obtainthe processing parameters for modeling the displacement caused byembossing (i.e., temperature 210, loading duration 230, and applied load240) by modeling the mechanical properties of the polymer and conductinga numerical simulation of the embossing process (assuming a set ofcandidate parameters).

FIG. 3 a is an illustration 300 of representation of global compressionof finite-thickness substrates before loading. FIG. 3 b is anillustration 301 of representation of global compression offinite-thickness substrates under uniform applied pressure in the zdirection.

Given that layer deflections and the characteristic dimensions of thefeatures being embossed are much smaller than the true thickness of thesubstrate 110, it is reasonable to assume that the shape of thetopography arising from spatial variation of the applied contactpressure will be substantially insensitive to the actual layerthickness. At the bottom of such a layer, the stress is almostindistinguishable from the stress that arises if a uniform contactpressure is applied at the top of the substrate. The amount of globalcompression of the substrate depends strongly on the original layerthickness and the average applied contact pressure. This globalcompression can be captured in the calculated value of Δ by adding aconstant term to every element of the filter g[m, n] or g[m].

Assuming that the substrate 110 is either infinite in extent andexperiences a periodic pressure distribution or laterally clamped at itsedges, the global compression of the substrate may be represented asshown in FIGS. 3A and 3B. Specifically, the applied embossing pressuremay be modeled as a uniform contact pressure distribution of magnitudep_(s,0), the infinite extent of the substrate in x and y directions or,equivalently, clamped substrate edges are reflected by the imposition ofzero strain in the x and y directions. Therefore, defining strains aspositive-compressive:

$\begin{matrix}{{{E\; ɛ_{zz}} = {p_{s,0} - {v\left( {p_{xx} + p_{yy}} \right)}}}{{E\; ɛ_{xx}} = {0 = {p_{xx} - {v\left( {p_{yy} + p_{s,0}} \right)}}}}{{E\; ɛ_{yy}} = {0 = {p_{yy} - {v\left( {p_{xx} + p_{s,0}} \right)}}}}{ɛ_{zz} = {\frac{p_{s,0}}{E}\left( {1 - \frac{2\; {v^{2}\left( {1 + v} \right)}}{1 - v^{2}}} \right)}}} & (1)\end{matrix}$

where the polymer network of the material deforms over time andapproaches a limiting configuration determined by the network's elasticmodulus denoted by E.

The convolution of a uniform contact pressure distribution of magnitudep_(s,0) with the filter g[m, n] implies the following displacement, Δ,of the substrate surface:

$\begin{matrix}{\Delta = {p_{s,0}{\sum\limits_{n = 0}^{N - 1}\; {\sum\limits_{m = 0}^{M - 1}{g\left\lbrack {m,n} \right\rbrack}}}}} & (2)\end{matrix}$

and Δ is proportional to the initial substrate thickness, h_(s):

Δ=∈₌h_(s)  (3)

such that:

$\begin{matrix}{{\sum\limits_{n = 0}^{N - 1}\; {\sum\limits_{m = 0}^{M - 1}{g\left\lbrack {m,n} \right\rbrack}}} = {\frac{h_{s}}{E}{\left( {1 - \frac{2\; {v^{2}\left( {1 + v} \right)}}{1 - v^{2}}} \right).}}} & (4)\end{matrix}$

The global compression behavior of the layer may therefore beapproximated by adding a constant value to every element of g[m, n] sothat it satisfies equation (4) above. The same approach works in theplane-strain case. The shape of the simulated topography is not affectedby this constant term in the filter and the only effect is a change inthe rigid-body stamp displacement Δ.

Accounting for Material Anisotropy

There are cases in which an embossed polymeric sheet is stronglyanisotropic. Thermoplastic sheets or plates are variously prepared bycasting, extrusion, and injection molding. In the cases of extrusion andinjection molding, polymer chains can become aligned in a particulardirection, and strongly directional residual stresses can be introducedby the process. Therefore, one aspect of the present invention capturespossible material anisotropy by making the point-load response of thematerial anisotropic. A new dimensionless parameter, k_(a), isintroduced to represent the strength of the anisotropy.

FIGS. 4 a-4 c illustrate effects of the anisotropy parameter anisotropicpoint-load response and its relation to the geometry of the embossedpattern. FIG. 4 a demonstrates a stamp design 400 defined in x 410 and y415 co-ordinates. FIG. 4 b illustrates the anisotropic point-loadresponse 420. The anisotropic point-load response 420 is a property ofthe material being embossed and is shown with reference co-ordinates e₁430 and e₂ 440. FIG. 4 c illustrates a 3-D rendering 450 of a typicalanisotropic point-load response. Specifically, the coordinate system ofthe material may be defined using the vectors (e₁, e₂) 430, 440 and thatof the stamp using (x, y) 410, 420. The lateral extent of the point-loadresponse 420 is shrunk by a factor of 1/k_(a) 450 in the material's e₂430 direction, relative to an isotropic material.

The anisotropic point load response may be spatially compressed by afactor 1/k_(a) in the e₂ direction, relative to the isotropic function.This compression captures the fact that an anisotropic material maypenetrate cavities oriented in a particular direction more easily thanthose oriented in other directions. When a long and narrow cavity isoriented parallel to the more elongated axis of the point load-responsefunction, the cavity will fill more deeply for a given applied pressurethan if the cavity were oriented perpendicular to the more elongatedaxis. The test pattern in FIG. 4 a includes cavities oriented in twoperpendicular directions, and so may be used to identify experimentallythe appropriate orientation of the anisotropic point-load response. Ifthe direction of anisotropy are at 45 degrees to the two trenchdirections, it may, however, be impossible to use this pattern todetermine the appropriate orientation of the point load-responsefunction. Therefore, a test pattern with trenches aligned in threeseparate directions may be used to allow for a more reliableidentification of the direction of any anisotropy.

The direction of orientation of polymer chains in the embossed materialmay correspond to the direction of the more elongated axis of theanisotropic point load response.

The material's anisotropy may be related to the coordinate system of thestamp 410, 420 by redefining the point-load response 420 of thesimulation's virtual elastic layer using:

$\begin{matrix}\left. \begin{matrix}{{g\left\lbrack {m,n} \right\rbrack} = {- {\frac{1 - v^{2}}{\pi \; E}\left\lbrack {{f\left( {x_{2},y_{2}} \right)} - {f\left( {x_{1},y_{2}} \right)} - {f\left( {x_{2},y_{1}} \right)} + {f\left( {x_{1},y_{1}} \right)}} \right\rbrack}}} \\{where} \\{{f\left( {x,y} \right)} = {{y\; {\ln \left( {x + \sqrt{x^{2} + y^{2}}} \right)}} + {x\; {\ln \left( {y + \sqrt{x^{2} + y^{2}}} \right)}}}} \\{and} \\\begin{matrix}{{x_{1} = {{md} - \frac{d}{2}}};{x_{2} = {{md} + \frac{d}{2}}};{y_{1} = {k_{a}\left( {{nd} - \frac{d}{2}} \right)}};} \\{{y_{2} = {{k_{a}\left( {{nd} + \frac{d}{2}} \right)}{for}\mspace{14mu} x{}e_{1}}};}\end{matrix} \\\begin{matrix}{{x_{1} = {k_{a}\left( {{md} - \frac{d}{2}} \right)}};{x_{2} = {k_{a}\left( {{md} + \frac{d}{2}} \right)}};} \\{{y_{1} = {{nd} - \frac{d}{2}}};{y_{2} = {{nd} + {\frac{d}{2}{for}\mspace{14mu} y{}e_{1}}}};}\end{matrix}\end{matrix} \right\} & (5)\end{matrix}$

Tuning the Modified Model to Represent Anisotropy in a Sample of Topas5013S

FIGS. 5 a-5 f are plots that illustrate measured peak cavity penetrationheights of the two samples' patterns embossed using aspects of thepresent invention.

Sample 1 (FIGS. 5 a-5 c) is generated using an injection-molded, 1mm-thick plate of the cyclic olefin polymer Topas 5013S that wasobtained as a gift from Topas Advanced Polymers (Florence, Ky.). Thisplate was cut into 25 mm-square pieces, and the orientation of eachpiece was noted relative to the framework of the supplied plate. One ofthe pieces was embossed with a silicon stamp carrying the test-patternof our earlier experiments. The relief of the test-pattern was 20 μm,and the x and y axes of the pattern were aligned to be parallel to theedges of the polymer piece. The embossing conditions were 95° C., 800kPa, and 4 minutes' loading duration. The material was cooled to 60° C.before unloading.

Sample 2 (FIGS. 5 d-5 f) is generated using a second square piece of theTopas 5013 material that was embossed under the same conditions, butwith the orientation of the polymer piece relative to the stamp patterndiffering by 90° from the previous experiment.

In both samples (samples 1 and 2), a strong orientation dependence ofthe depth of penetration of the long, slender rectangular cavities isobserved. In Sample 1, the rectangular cavities 510 whose long axis runsparallel to the x-axis of the stamp have filled much less deeply thanthe cavities 520 running parallel to the y-axis. In Sample 2, this trendis reversed, with the cavities running parallel to the stamp's x-axispenetrating more deeply. This strong change in behavior indicates thatthe principal axes of the material's anisotropy run approximatelyparallel with the edges of the polymer squares embossed.

The anisotropy of this sample of Topas 5013S, without calibrating a fullviscoelastic material model may be modeled by fitting a value for k_(a),and a value for the effective compliance of the material given thefour-minute, 95° C. processing conditions used. A reasonably closeapproximation to experimental results may be obtained by taking a valueof 4 for k_(a) and a value of 1.6 MPa for the effective stiffness, E, ofthe polymeric substrate, as defined in equation (5). Poisson's ratio, v,is assumed to be 0.5. The output of a simulation assuming theseparameters is shown by the plotted lines 530, 540, 550 in FIG. 5 a-5 f.

By modeling material anisotropy, embodiments of the present inventionare able to detect the material's anisotropy. Certain embodiments mayenable a test pattern to detect anisotropy by including slenderrectangular cavities having three or more orientations.

The extent of material anisotropy of any given polymer plate is a strongfunction of its processing history. The anisotropy of one particularbatch of a particular material is therefore not necessarilyrepresentative of any other batch of the same material. Accordingly,embossing tests may be needed to identify batch-to-batch variation ofmaterial anisotropy.

Although the approach developed above assumes a Voigt material model torepresent the micro-embossing behavior of the relatively highmolecular-weight polymers, alternative representations of the polymermay be used.

Generalizing Compliance Functions

Aspects of the present invention may be adapted to use any linearfunction of time to represent “compliance” of the polymer. The termcompliance refers to a characteristic parameter that relates to the meanvalue of the peak penetration depths z_(pk) across all regions of thetest pattern, divided by the average applied embossing pressure p₀.

If the compliance is denoted by J(t), the expression for the response ofthe surface of a half-space to a point load applied for all time t>0becomes:

$\begin{matrix}{{s_{general}\left( {x,y,t} \right)} = \frac{{- \left( {1 - v^{2}} \right)}{J(t)}}{\pi \sqrt{x^{2} + y^{2}}}} & (6)\end{matrix}$

and the corresponding impulse response becomes:

$\begin{matrix}{{g_{general}\left( {x,y,t} \right)} = {\frac{{- \left( {1 - v^{2}} \right)}\frac{{J(t)}}{t}}{\pi \sqrt{x^{2} + y^{2}}}.}} & (7)\end{matrix}$

The convolution integral describing the embossed topography becomes:

$\begin{matrix}{{w\left( {x,y,t} \right)} = {\frac{- \left( {1 - v^{2}} \right)}{\pi}{\int_{0}^{t}{\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{\frac{{p\left( {x^{\prime},y^{\prime},t^{\prime}} \right)}\frac{{J\left( {t - t^{\prime}} \right)}}{t^{\prime}}}{\sqrt{\left( {x - x^{\prime}} \right)^{2} + \left( {y - y^{\prime}} \right)^{2}}}\ {x^{\prime}}\ {y^{\prime}}\ {t^{\prime}}}}}}}} & (8)\end{matrix}$

and weighted time-average of applied pressure is:

$\begin{matrix}{{p_{g}\left( {x,y,t_{h}} \right)} = {\left( {1 - v^{2}} \right){\int_{0}^{t_{h}}{{p\left( {x,y,t^{\prime}} \right)}\ \frac{{J\left( {t - t^{\prime}} \right)}}{t^{\prime}}{{t^{\prime}}.}}}}} & (9)\end{matrix}$

Averaging spatially over one period of the embossed pattern:

$\begin{matrix}{{p_{g,0}\left( t_{h} \right)} = {{\left( {1 - v^{2}} \right){\int_{0}^{t_{h}}{\frac{1}{D^{2}}\ {\int_{0}^{D}{\int_{0}^{D}{{p\left( {x,y,t^{\prime}} \right)}\ {x}\ {y}\frac{{J\left( {t - t^{\prime}} \right)}}{t^{\prime}}{t^{\prime}}}}}}}} = {\left( {1 - v^{2}} \right){\int_{0}^{t_{h}}{{p_{0}\left( t^{\prime} \right)}\ \frac{{J\left( {t - t^{\prime}} \right)}}{t^{\prime}}{{t^{\prime}}.}}}}}} & (10)\end{matrix}$

From this, equation (8) may be rewritten as:

$\begin{matrix}{{w\left( {x,y,t_{h}} \right)} = {\frac{- 1}{\pi}{\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{\frac{p_{g}\left( {x,y,t_{h}} \right)}{\sqrt{\left( {x - x^{\prime}} \right)^{2} + \left( {y - y^{\prime}} \right)^{2}}}{x^{\prime}}{{y^{\prime}}.}}}}}} & (11)\end{matrix}$

Expression (11) is equivalent to instantaneously embossing an elasticsubstrate with E/(1−v²)=1, with the dimensionless pressure distributionp_(g)(x, y, t_(h)). From expression (10), the spatial average ofp_(g)(x, y, t_(h)) as a function of the spatial average of true appliedpressure over time is known. Therefore, w(x, y, t_(h)) may be foundusing the spatial average.

FIG. 6 is a table that includes an outline of some possible polymermodels and their corresponding compliance functions J(t). Specifically,purely elastic 610, Newtonian liquid 620, Maxwell model 630,Kelvin-Voigt Model 640, and hybrid Maxwell model with Voigt Model 650,possible uses of each polymer, their corresponding compliance functionsJ(t), and their corresponding change over time are outlined.

Nanoimprint Lithography

Certain aspects of the present invention relate to modeling nanoimprintlithography. In nanoimprint lithography, cavity widths are oftencomparable to layer thicknesses. Nanoimprint lithography may beassociated with several modeling challenges. For example, a layer ofmaterial being nanoimprinted is usually reduced to a small proportion ofits initial thickness in certain regions of the pattern, and it will beessential to represent the increasing difficulty of reducing thisresidual layer's thickness as it becomes progressively thinner. Further,in nanoimprint lithography, the deflections of the stamp and the hardsubstrate underlying the imprinted layer are often of crucialimportance. While the deflections of a half millimeter-thick siliconstamp when compared with the deflections of a softened millimeter-thickplate of PMMA are negligible, in nanoimprint lithography, thesedeflections become comparable in magnitude to the deformations of thepolymeric resist layer, which might be only 100 nm thick.

Also, in nanoimprint lithography, in addition to ensuring that everycavity on a stamp is filled with resist material during a particularimprint process, it is often required to determine the thickness anduniformity of the residual layer of imprinted resist since this residuallayer is often etched through in a post-imprint fabrication step and anyvariation in layer thickness imposes a need for over-etching (i.e.,prolonging the duration of the etching step to ensure that even thethickest parts of the residual layer are penetrated). Residual layerthickness variation propagates to variation of the critical dimensionsof the underlying etched features, and probably to unwanted deviceperformance variation. Therefore, it is desirable to be able to control,or predict, residual layer thickness variation for a given imprintingpattern and process.

Various nanoimprint machine designs may be employed. For example,wafer-scale imprinting of spun-on thermoplastic or thermosetting layers,‘Step-and-flash’ imprint (stepping with a die-sized stamp across thewafer and filling the cavities with freshly sprayed-on liquid resistbefore being cured with UV light), and Roll-to-roll nanoimprinting maybe employed. Wafer-scale embossing of a spun-on visco-elastic layer isdescribed herein. Wafer-scale embossing is a widely-used process and isdeployed in commercially available imprint tools and in academicresearch. An additional, realistic, assumption that may be made is thatthe wafer is patterned with a square array of nominally identical dice,such that the pattern is approximated as being periodic in space.Further, the relationships between the time-course of applied pressure,visco-elastic layer parameters, and imprinted layer topographies areparameterized for three simple types of imprinted geometry: parallellines, square holes, and circular pads. These relationships areintegrated with a description of stamp and substrate deflections toconstruct a technique for simulating the imprinting of feature-richpatterns.

Modeling Stamp and Substrate Deflections

The imprinting system being modeled include polymer layers that sit on asubstrate which is relatively hard and perfectly elastic, and which maybe regarded as a half-space consisting of the wafer on to which theresist has been spun and the underlying platen (made, for example, ofsteel). Meanwhile, a patterned elastic stamp of finite thickness ispressed hydrostatically into the resist. The use of a hydrostaticpressure to press the stamp against the imprinted layer (the so-called‘air cushion’ press design [4-5]) is intended to counteract wafer-scalenon-uniformity by allowing the stamp to bend and conform to long-rangenon-flatness of the stamp or substrate.

To model the deflections of stamp and substrate within our framework,certain embodiments require that the expressions for the responses ofthe stamp and substrate surfaces to unit contact pressure to be applieduniformly across one d×d region.

Further, any compression of individual feature protrusions from thestamp may be ignored, since their heights are a tiny fraction of thestamp thickness, and they are in almost all cases much harder than theresist itself, individual stamp feature compression can usually besafely neglected.

Substrate Deflections

The deflections of the substrate may be modeled as:

$\begin{matrix}\left. \begin{matrix}{{g_{substrate}\left\lbrack {m,n} \right\rbrack} = {\frac{1 - v_{substrate}^{2}}{\pi \; E_{substrate}}\begin{bmatrix}{{f\left( {x_{2},y_{2}} \right)} - {f\left( {x_{1},y_{2}} \right)} -} \\{{f\left( {x_{2},y_{1}} \right)} + {f\left( {x_{1},y_{1}} \right)}}\end{bmatrix}}} \\{where} \\{{f\left( {x,y} \right)} = {{y\; {\ln \left( {x + \sqrt{x^{2} + y^{2}}} \right)}} + {x\; {\ln \left( {y + \sqrt{x^{2} + y^{2}}} \right)}}}} \\{and} \\{{x_{1} = {{md} - \frac{d}{2}}};{x_{2} = {{md} + \frac{d}{2}}};{y_{1} = {{nd} - \frac{d}{2}}};{y_{2} = {{nd} + {\frac{d}{2}.}}}}\end{matrix} \right\} & (12)\end{matrix}$

Appropriate values for E_(substrate) and v_(substrate) may be those ofsilicon: ˜120 GPa and 0.27 respectively.

Stamp Deflections

The deflections of the stamp, meanwhile, may be regarded as having twocomponents: an ‘indentation’ component, and a ‘bending’ component. Theindentation component dominates cases where the characteristic spatialperiod of the contact pressure distribution is smaller than thethickness of the stamp. A good approximation for the kernel functiondescribing the indentation component, g_(stamp,indentation)[m, n], hasthe same form for substrate indentations. The values of E and v may bethose of the stamp material; since stamps are often made from silicon,g_(stamp,indentation)[m, n] and g_(substrate)[m, n] will frequently beidentical in magnitude.

The stamp's bending mode, in contrast, may be expected to dominatedeflections, where the characteristic spatial period of the contactpressure distribution is larger than approximately the stamp thickness.While most patterns imprinted using NIL have lateral feature separationsthat are far smaller than the stamp thickness, there may be patterndensity variations that occur over distances greater than a stampthickness, so we do wish to allow for the possibility of stamp bending.The stamp bending kernel may be derived by exploiting the assumption ofspatial periodicity. The application of unit stamp-substrate contactpressure across a given d×d region of the stamp is balanced by ahydrostatic pressure of d²/D² applied to the backside of the stamp,assuming that the spatial period of the stamp is D in each direction.Because of the assumption of spatial periodicity, the application ofunit pressure in one d×d region of the simulated pattern is equivalentto the application of unit pressure at an infinite array of d×d regions,spaced D apart in both directions across an infinitely large stamp.Looking at one particular D×D period of this imagined infinite stamp,the stamp's deformed shape is identical to that of an edge-clampedsquare plate of size D×D, experiencing uniform pressure of d²/D² on oneside and unit pressure applied on the other side to a d×d region at itscenter. The deflections of this equivalent clamped plate can be foundusing the standard analytical results of Timoshenko [6] and Reddy [7].The following two plate topographies may be superimposed: (i) g_(U)(x,y), that resulting from the hydrostatic applied pressure of d²/D², and(ii) g_(P)(x, y), that resulting from the localized load at the centerof the plate:

g _(stamp,bending)(x,y)=g _(U)(x,y)−g _(P)(x,y)  (13)

The localized load is modeled as a point load of magnitude d² to enablethe ready use of standard formulae. Following the methods used byTimoshenko and Reddy, g_(U)(x, y) and g_(P)(x, y) are each given by thesuperposition of two sets of deflections: the deflections g_({U|P},s)(x,y) of a simply supported plate exposed to the loads just described, andthe deflections g_({U|P},m)(x, y) in response to edge clamping momentsthat satisfy the condition of no edge rotations of the plate. Thefunction g_({U|P},m)(x, y) describes the deflections resulting fromclamping moments applied to only two opposite edges of the plate's fouredges, whereas in fact clamping moments are experienced at all fouredges so that the overall deflections g_({U|P})(x, y) include bothg_({U|P},m)(x, y) and its transpose, g_({U|P},m)(y, x).

$\begin{matrix}\left. \begin{matrix}{{g_{U}\left( {x,y} \right)} = {{g_{U,s}\left( {x,y} \right)} + {g_{U,m}\left( {x,y} \right)} + {g_{U,m}\left( {y,x} \right)}}} \\{{g_{P}\left( {x,y} \right)} = {{g_{P,s}\left( {x,y} \right)} + {g_{P,m}\left( {x,y} \right)} + {g_{P,m}\left( {y,x} \right)}}}\end{matrix} \right\} & (14)\end{matrix}$

FIG. 7 illustrates Kernel functions used for describing stamp bending incertain aspects of the present invention. The discretized bending kernelfor the stamp, g_(stamp,bending)[m, n], is a discrete representation ofg_(plate)(x, y) that is obtained by sampling g_(plate)(x, y) sampled atspatial intervals of d.

Expressions for g_({U|P},m)(x, y) and g_({U|P},s)(x, y) are given below:

$\begin{matrix}{{w_{U,s}\left( {x,y} \right)} = {\frac{4\; D^{4}}{N^{2}\pi^{5}P}{\sum\limits_{{m = 1},3,5,7}\; {\frac{\left( {- 1} \right)^{\frac{m - 1}{2}}}{m^{5}}{{\cos \left( \frac{m\; \pi \; x}{D} \right)}\left\lbrack \left. \quad{1 - {\frac{{\alpha_{m}\tan \; h\; \alpha_{m}} + 2}{2\; \cosh \; \alpha_{m}}{\cosh \left( \frac{m\; \pi \; y}{D} \right)}} + {\frac{m\; \pi}{2\; D\; \cosh \; \alpha_{m}}y\; \sin \; h\frac{m\; \pi \; y}{D}}} \right\rbrack \right.}}}}} & (15) \\{{w_{U,m}\left( {x,y} \right)} = {{- \frac{2\; D^{4}}{N^{2}\pi^{5}P}}{\sum\limits_{{m = 1},3,5,7}{\frac{{Q_{m}\left( {- 1} \right)}^{\frac{m - 1}{2}}}{m^{2}\cosh \; \alpha_{m}}{{\cos \left( \frac{m\; \pi \; x}{D} \right)}\left\lbrack {{\frac{m\; \pi}{D}y\; {\sinh \left( \frac{m\; \pi \; y}{D} \right)}} - {\alpha_{m}\tanh \; \alpha_{m}{\cosh \left( \frac{m\; \pi \; y}{D} \right)}}} \right\rbrack}}}}} & (16) \\{{w_{P,s}\left( {x,y} \right)} = {\frac{4\; d^{2}}{D^{2}P}{\sum\limits_{n = 1}^{7}\; {\sum\limits_{m = 1}^{7}{\frac{1}{k_{mn}}{\sin \left( \frac{m\; \pi}{2} \right)}{\sin \left( \frac{n\; \pi}{2} \right)}{\sin\left( \frac{m\; {\pi \left( {x - \frac{D}{2}} \right)}}{D} \right)}{\sin\left( \frac{n\; {\pi \left( {y - \frac{D}{2}} \right)}}{D} \right)}}}}}} & (17) \\{\mspace{79mu} {k_{mn} = {\left( \frac{\pi}{D} \right)^{4}\left( {m^{4} + {2\; m^{2}n^{2}} + n^{4}} \right)}}} & (18) \\{{w_{P,m}\left( {x,y} \right)} = {{- \frac{D^{2}d^{2}}{2\pi^{2}P}}{\sum\limits_{{m = 1},3,5,7}{\frac{{R_{m}\left( {- 1} \right)}^{\frac{m - 1}{2}}}{m^{2}\cosh \; \alpha_{m}}{{\cos \left( \frac{m\; \pi \; x}{D} \right)}\left\lbrack {{\frac{m\; \pi}{D}y\; {\sinh \left( \frac{m\; \pi \; y}{D} \right)}} - {\alpha_{m}\tanh \; \alpha_{m}{\cosh \left( \frac{m\; \pi \; y}{D} \right)}}} \right\rbrack}}}}} & (19) \\{\mspace{79mu} {\alpha_{m} = \frac{m\; \pi}{2}}} & (20) \\{\mspace{79mu} {P = \frac{{Et}_{plate}^{3}}{12\left( {1 - v^{2}} \right)}}} & (21)\end{matrix}$

In equations (15) and (16), N=D/d. The values of Q_(m) and R_(m)referred to in (16) and (19) respectively are given by Timoshenko. FIG.8 is a table that includes values of Q_(m) and R_(m) required forconstruction of the plate bending function Kernel. As shown in FIG. 8,the introduction of the m=7 terms makes no appreciable difference to theshapes of the final kernels, compared to the case of using all lowerterms but not m=7 itself. It is therefore concluded unnecessary to addterms above m=7.

The plate-bending expressions derived are for a uniform-thickness,isotropic plate. However, other expressions may be derived for layeredor composite plates in a similar manner. Further, the expressions forthe kernel functions may be generalized to deal with anisotropicmaterials in a similar manner.

Quantifying the Relative Magnitudes of Stamp Indentation and Bending

Stamp indentation may dominate deflections when the characteristicspatial period of the imprinted pattern is lower than approximately astamp thickness. Further, bending may dominate for larger spatialperiods. Certain aspects of the present invention quantify the relativeimportance of these two modes through simulations that use the kernelfunctions derived above.

Hard surfaces of topography z(x, y) are defined as follows, anddiscretized:

$\begin{matrix}{{z\left( {x,y} \right)} = {z_{0}{\sin \left( {2\pi \frac{x}{\lambda}} \right)}{{\sin \left( {2\pi \frac{y}{\lambda}} \right)}.}}} & (22)\end{matrix}$

Here, λ represents the characteristic spatial period of the topography.The simulations may be performed such that an initially flat layer ofmaterial having the elastic properties of the stamp is forced to conformto this hard topography. Firstly, only the indentation mode of the stampis considered. For each of a range of values of λ, the contact pressuredistribution required to pull the stamp fully into contact with thesurface z(x, y) may be determined. The pressure distributions are foundby de-convolving the discretized z[m, n] with org_(stamp,indentation)[m, n]. Arbitrary values may be used for z₀ and forthe stamp's plate modulus, E′. For each calculation, the amplitude ofthe pressure distribution, p_(z0,i), is recorded.

The procedure may be further repeated by considering only the bendingmode of the stamp. The pressure distributions may be found byde-convolving the discretized z[m, n] with g_(stamp,bending)[m, n]. Anarbitrary value may be adopted for the stamp thickness, t_(stamp). Foreach calculation the amplitude of the pressure distribution, p_(z0,b),is recorded.

It is noted that p_(z0,i) is inversely proportional to the wavelength ofthe surface, while p_(z0,b) is inversely proportional to the fourthpower of the wavelength. The constants k_(i) and k_(b) are then definedas follows:

$\begin{matrix}{{p_{{z\; 0},i} = \frac{k_{i}z_{0}E^{\prime}}{\lambda}};} & (23) \\{{p_{{z\; 0},b} = \frac{k_{b}z_{0}E^{\prime}t_{stamp}^{3}}{\lambda^{4}}};} & (24) \\{E^{\prime} = {\frac{E}{1 - v^{2}}.}} & (25)\end{matrix}$

The values of these constants may be extracted from the numericalsimulation. For example, k_(i) may be found to be 4.28 and k_(b) to be336 (3 s.f.). By defining the value of λ at which p_(z0,b)=p_(z0,i) tobe that at which bending and indentation are of equal importance, stampbending may be found to begin to play a larger role than bulkindentation in conforming to the surface when

$\begin{matrix}{\frac{\lambda}{t_{stamp}} > \sqrt[3]{\frac{k_{b}}{k_{i}}} \approx 4.} & (26)\end{matrix}$

Since stamp compliance is what enables residual layer non-uniformity toarise, it may seem desirable, from the perspective of minimizing patterndependencies, to make stamps as stiff as possible. One good reason fornot wishing to make a stamp infinitely stiff, however, is that stamps,substrates, and polymeric layers may not be perfectly flat at the startof the imprinting process. For example, they may be wavy, or bowed, orhave a certain roughness, or have dust particles sitting on theirsurface. A stamp with a certain amount of compliance may conform tothese imperfections and reduce the lateral extent of their influence;that indeed is the purpose of the air cushion press design.

As a proxy for the ability of a stamp to conform to layer topographyimperfections, certain embodiments may use the level of contact pressureuniformity that arises when an elastic layer having the properties ofthe stamp is pressed against a hard topography representing the unevenlayer. Suppose that a specification is placed on the pressurenon-uniformity arising from layer waviness of a certain wavelength, suchthat it must not exceed a fraction α of the average contact pressure p₀.

FIG. 9 is an illustration 900 of contact pressure non-uniformity for ahard rough surface and a stamp that can either bend 920 or be indented910. For an incompressible layer material 930, the total contactpressure distribution is equal to the zero-mean pressure distributionrequired to just bring the elastic layer into contact with all points ofthe topography, plus a constant p₀.

Therefore, at the wavelength (shown as plot 940) at which stamp bendingbecomes more significant than indentation, the largest allowableroughness amplitude z₀ that will keep the pressure within itsnon-uniformity specification may be given by:

$\begin{matrix}{\frac{z_{0}}{t_{stamp}} = {\sqrt[3]{\frac{k_{b}}{k_{i}}}{\left( \frac{\alpha \; P_{0}}{E^{\prime}k_{i}} \right).}}} & (27)\end{matrix}$

FIG. 10 is a schematic graph 1000, on logarithmic axes, of the maximumsurface waviness to which a stamp can conform as a function of theapplied average pressure, the pressure non-uniformity specification, thematerial properties, and the characteristic wavelength of surfaceroughness. As shown in FIG. 10, the maximum allowable surface wavinessamplitude may be plotted against the characteristic wavelength of thelayer imperfections. Near the transition region, both indentation 1010and bending 1020 are in fact relevant, and the dashed line in FIG. 10suggests how the maximum acceptable z₀ might, in reality, change with λ.

Modeling the Use of Layered Stamps

Given the dual aims, when designing a stamp, of minimizingpattern-dependent residual layer non-uniformity and accommodating anyinitial layer imperfections, one may imagine that it would be useful totailor the wavelength, compliance spectrum of the stamp, i.e., the shapeof the curve illustrated in FIG. 10. For example, stamps composed of twoof more layers of varying stiffness, or stamps with graded,depth-varying stiffness, may provide better performance than a single,thick elastic layer.

If, for example, polymer layers of a certain type are known to exhibitbow or waviness but very little roughness below a particularcharacteristic wavelength, it may be valuable to produce a stamp inwhich the features were patterned on a relatively stiff surface layerattached to a more compliant support layer. This way, initial layerwaviness may be accommodated while residual layer non-uniformity issuppressed at length scales smaller than that of the waviness. Such anapproach has been demonstrated by Suh et al. [8].

Conversely, if the layer being imprinted is particularly rough at smalllength-scales, it may be beneficial to make the stamp with a softsurface layer on a harder support material. McClelland et al. report theuse of this approach for the patterning of resist spun on to roughmagnetic layers [9].

Certain embodiments tailor the compliance of the stamp so that the shapeof its wavelength−compliance (z₀−λ) spectrum hugs as closely as possiblethe characteristic spectrum of any parasitic roughness/waviness of thestamp and substrate. This way, the stamp does not deform any more thanis needed to counteract these parasitic topographies, and sopattern-dependent residual layer variation is no larger than it needs tobe. In fact, there is likely to be a trade-off between accommodating anyparasitic roughness/waviness of the stamp and substrate and limitingpattern-dependent stamp deflections.

Certain embodiments of the invention employ a two-layer stamp modelingas a way of tailoring stamp compliance. FIG. 11 illustrates the factorsdetermining the wavelength−compliance spectrum of a stamp with a coatinglayer of thickness t_(layer) and elasticity E₁ attached to a thickbacking layer of stiffness E₂. The kernel function described by Nogi andKato [2] for the point load response of a layered elastic substrate areused in the numerical simulations that yielded these results. WhereE₁<E₂, stamp compliance is large for λ<<t_(layer), exhibits a plateauaround λ=t_(layer), and begins increasing again for λ>>t_(layer),corresponding to substantial indentation of the backing layer. WhereE₁>E₂, the thin layer exhibits indentation behavior for λ<<t_(layer),behaves as a plate on an elastic support for λ=t_(layer), and transmitsloads fairly directly to the soft support layer for λ>>t_(layer).

By tuning the thickness of the coating layer and the elasticity of bothlayers, it may therefore be possible to ‘hug’ the parasiticroughness/waviness spectrum of a stamp and substrate more closely thancan be done with a single-layer substrate. Multi-layer orgraded-stiffness substrates could offer even better tailoring thantwo-layer stamps, although the analysis would be substantially morecomplicated and is left for the future.

Certain embodiments may tailor stamp compliance in other ways. Forexample, backside grooves may be introduced into nanoimprint stamps,usually aligned with the edges of dice [10-12]. These grooves increasethe bending compliance of the stamp at length scales greater than thatof the die size, and are claimed to enhance wafer-scale imprintuniformity. In step-and-flash imprint lithography, meanwhile, becausethe stamp is the size of a single die, it is possible to reducewafer-scale non-uniformity by using mechanical systems that maneuver thestamp to be parallel to the local surface of the wafer before eachimprint is made [13].

If it is inconvenient to tailor the stamp compliance, it may be possibleto improve conformality of stamp and substrate by placing a compliantlayer beneath the substrate itself. This approach is applicable forcounteracting stamp or substrate waviness at length scales larger thanthe substrate thickness.

Hierarchical Approach

Certain embodiments bring together the models for polymer layerdeformation and stamp/substrate deflections to construct a die-scale,hierarchical simulation method. As described above, each (identical) D×Ddie on the stamp may be discretized into regions of size d×d. The diedesign is described by a ‘coarse’ topography w_(stamp)[m, n] that takesa uniform value within any given d×d region, upon which is superimposeda ‘fine’ topography characterized within each d×d region by ahomogeneous, regular pattern of a particular pitch, shape, and arealdensity. In this way, a single simulation approach may be used both forlarge, feature-rich patterns and for intricate custom patternsdiscretized at the sub-feature scale.

Since the model considers a large square array of identical dice, anywafer-scale components of stamp and layer deformation may be ignored.Assuming that wafer-scale variation only becomes significant when allcavities within the die have been substantially filled, the followingkinematic relationship between the reference displacement of the stamp,Δ, and the deflections of stamp, substrate and polymer layer inside thestamp−layer contact region C may be developed:

w _(stamp) [m,n]+w _(D,die) [m,n]+w _(D,substrate) [m,n]+w _(local)[m,n]+w _(die) [m,n]=Δ∀m,n∈C  (28)

The reference displacement of the stamp, Δ, is definedpositive-downwards into the polymeric layer. The coarse stamp topographyw_(stamp)[m, n] is fixed for a given stamp and is defined positiveupwards from the surface of the stamp that is in contact with thepolymeric layer at the start of imprinting. Intra-die deflections of thesurface of the stamp, w_(D,die), are defined positive-upwards.Deflections of the substrate, w_(D,substrate), are definedpositive-downwards. Both w_(D,die) and w_(D,substrate) are assumed hereto be periodic at the die level and to have no wafer-scale variation.Because the substrate is assumed elastic, substrate deflectionsconsistent with the final imprinting pressure distribution spring backwhen the stamp is removed.

Deformations of the polymeric layer are defined such that the residuallayer thickness r[m, n]—i.e. the thickness of the thinnest part ofmaterial within that region is given by the following relation:

r[m,n]=r ₀ −w _(local,u) [m,n]−w _(die,u) [m,n]  (29)

where r₀ is the initial layer thickness. The quantity w_(local)[m, n]represents the reduction of residual thickness that is associated withthe filling of any fine features within the region [m, n]. Meanwhile,w_(die)[m, n] represents any reduction of layer thickness that isconsistent with the exchange of layer material between adjacent d×dregions. The quantity w_(die)[m, n] is therefore responsible forcapturing intra-die pattern interactions. The objective is to find thevalues of three particular quantities at the end of the imprintingcycle: (i) the residual layer thickness everywhere in the contactregion, C; (ii) the proportions of the volumes of cavities constitutingthe ‘fine’ topography that are filled with polymer; (iii) the topographyof material outside the contact region. The solution for these threequantities are obtained in terms of the dimensionless contact pressurecompliance, p_(g)[m, n]. The simulation procedure involves graduallyincreasing this quantity's spatial average, p_(g,0), in a series ofsteps, labeled u=1 . . . U, towards the final value p_(g,0,U). Thisfinal value is a function of the linear viscoelastic properties of theimprinted layer and the pressure−time profile of the imprinting processbeing simulated.

This stepping procedure serves to deal with the geometric nonlinearitiesthat exist in the imprinting of a finite-thickness layer. At each step,the ‘resistance’ to imprinting is recomputed as a function of position,according to the instantaneous maps of r[m, n] and p_(g)[m, n] obtainedin the previous step.

If the resist model is a Newtonian viscosity and the applied averagepressure remains constant with time, then the value of p_(g,0) at agiven step is proportional to the imprinting time that has elapsed. Inthe cases of different viscoelastic models or different pressure−timecourses, the value of p_(g,0) is a function of both elapsed time andapplied pressure history.

FIG. 12 is a table that includes initial values (i.e., u=1) of variablesinvolved in nanoimprint simulation. For subsequent steps u=2, . . . , U,the following procedure is followed:

Compute g_(die,u-1)[m, n]

A kernel g_(die,u-1)[m, n] is generated for an elastic layer on aperfectly hard substrate, using the expressions presented by Nogi andKato [2] and in the previous chapter. The thickness of the layer is setto the mean value of r_(u-1)[m, n]. The kernel is generated on a lateralgrid of pitch d. Spatial variation of the residual layer thickness isencapsulated in the quantity k_(die,u-1)[m, n], discussed below. Becausewe are working in terms of the dimensionless pressure−compliancequantity p_(g)[m, n], we generate the kernel assuming a dimensionlessYoung's modulus of (1−v²) and a Poisson's ratio of v=0.5.

Compute k_(inst,u-1)[m, n]

The value of k_(inst,u-1)[m, n] at each step is defined as:

$\begin{matrix}{{k_{{inst},{u - 1}}\left\lbrack {m,n} \right\rbrack} = \left. {\frac{- {w_{local}}}{p_{g}}\left\lbrack {m,n} \right\rbrack} \right|_{r = r_{u - 1}}} & (30)\end{matrix}$

If S_(u-1)[m, n] equals 1 (i.e. the region is in squeezing-dominatedmode), then k_(inst,u-1)[m, n] is calculated using the appropriatelyscaled dp_(g)/dr′ as defined in

${- \frac{p_{g,0}}{\rho {r^{\prime}}}} = {{F_{1}\left( r^{\prime} \right)}^{- 3} + {F_{2}\left( r^{\prime} \right)}^{- 2} + {F_{3}.}}$

If S_(u-1)[m, n] equals 0 (i.e. the region is in bulkdeformation-dominated mode), then k_(inst,u-1)[m, n] is calculated usingthe appropriately scaled derivative of V/V₀[m, n] with respect top_(g)[m, n].Solve for p_(g,u)[m, n] and the Corresponding Value of Δ_(u)

At each step, we have specified the spatial-average of the virtualpressure p_(g,0,u). We decompose the solution for p_(g,u)[m, n] into twoparts:

$\begin{matrix}{{{{p_{g,u}^{f}\left\lbrack {m,n} \right\rbrack}*\left\{ {\frac{p_{0}\left( t_{k} \right)}{p_{g,0,u}}{g_{ss}\left\lbrack {m,n} \right\rbrack}} \right\}} + {\left\{ {{k_{{die},{u - 1}}\left\lbrack {m,n} \right\rbrack}{p_{g,u}^{f}\left\lbrack {m,n} \right\rbrack}} \right\}*{g_{{die},{u - 1}}\left\lbrack {m,n} \right\rbrack}} - {{p_{g,u}^{f}\left\lbrack {m,n} \right\}}{k_{{inst},{u - 1}}\left\lbrack {m,n} \right\rbrack}}} = {{- {w_{stamp}\left\lbrack {m,n} \right\rbrack}} - r_{0} + {r_{u - 1}\left\lbrack {m,n} \right\rbrack} - {{p_{g,{u - 1}}\left\lbrack {m,n} \right\rbrack}{k_{{inst},{u -}}\left\lbrack {m,n} \right\rbrack}} + {{g_{{die},{u - 1}}\left\lbrack {m,n} \right\rbrack}*\left( {k_{{die},{u - 1}}{p_{g,{u - 1}}\left\lbrack {m,n} \right\rbrack}} \right)}}} & (31) \\{{{{p_{g,u}^{e}\left\lbrack {m,n} \right\rbrack}*\left\{ {\frac{p_{0}\left( t_{h} \right)}{p_{g,o,u}}{g_{ss}\left\lbrack {m,n} \right\rbrack}} \right\}} + {\left\{ {{k_{{die},{u - 1}}\left\lbrack {m,n} \right\rbrack}{p_{g,u}^{e}\left\lbrack {m,n} \right\rbrack}} \right\}*{g_{{die},{u - 1}}\left\lbrack {m,n} \right\rbrack}} - {{p_{g,u}^{e}\left\lbrack {m,n} \right\rbrack}{k_{{inst},{u - 1}}\left\lbrack {m,n} \right\rbrack}}} = 1} & (32)\end{matrix}$

We have the additional constraint that contact pressure is zero outsideC:

p_(g,u)[m,n]=0∀m,n∉C.  (33)

Pressure distributions p^(f) _(g,u)[m, n] and p^(e) _(g,u)[m, n] may befound using the stabilized biconjugate gradient algorithm. Having foundthese two distributions, we solve for the stamp reference displacementΔ_(u) such that:

p ^(e) _(g,u) [m,n]+Δ _(u) p ^(f) _(g,u) [m,n]=p _(g,u) [m,n]  (34)

subject to the constraint that:

$\begin{matrix}{{\sum\limits_{m = 1}^{M}{\sum\limits_{n = 1}^{N}{p_{g,u}\left\lbrack {m,n} \right\rbrack}}} = {{MNp}_{g,0,u}.}} & (35)\end{matrix}$

Equations (31) and (32) encapsulate the idea that in each simulationstep the layer deformation at each location [m, n] may experience changefrom two sources: (i) the forcing of material from the residual layerinto the finely patterned cavities within the region [m, n], as governedby the ‘stiffness’ coefficient k_(inst,u-1)[m, n], and (ii) thedisplacement of material laterally between adjacent regions, as governedby g_(die,u-1)[m, n].

The use of a single kernel, g_(die,u-1)[m, n], to represent the lateraltransport of material between regions allows the procedure for findingp_(g,u)[m, n] to consist of a simple series of convolutions andmultiplications, but does not in itself capture the fact that itrequires a larger pressure to force material away from a region withthinner r_(u-1)[m, n] than from a region in which r_(u-1)[m, n] isthicker. The pre-multiplication factor k_(die,u-1)[m, n] is designed toreflect any spatial variation of r_(u-1)[m, n]: in (31) and (32) it ismultiplied by the trial pressure distribution p_(g,u)[m, n] beforeconvolution with g_(die,u-1)[m, n]. In this way, residual layerthickness variation may be approximately accounted for without slowingdown the solution procedure. In layers where r_(u-1)[m, n] is everywheresubstantially smaller than the lateral pitch d at which g_(die,u-1)[m,n] is defined, k_(die,u-1)[m, n] may be proportional to the cube ofr_(u-1)[m, n].

In the convolution expressions of (31) and (32), the kernel g_(ss)[m, n]incorporates both substrate and stamp deflections: since all substratedeflections spring back upon unloading, it is convenient to fold thesubstrate and stamp deformations into a single kernel. The individualkernels for bending and indentation are as defined above.

g _(ss) [m,n]=g _(stamp,bending) [m,n]+g _(stamp,indentation) [m,n]+g_(substrate) [m,n]  (36)

The sum of stamp and substrate deflections is then given by thefollowing convolution:

$\begin{matrix}{{{w_{D,{die},u}\left\lbrack {m,n} \right\rbrack} + {w_{D,{substrate},u}\left\lbrack {m,n} \right\rbrack}} = {\frac{p_{0}\left( t_{h} \right)}{p_{g,0,u}}{p_{g,u}\left\lbrack {m,n} \right\rbrack}*{{g_{ss}\left\lbrack {m,n} \right\rbrack}.}}} & (37)\end{matrix}$

As seen in (31) and (32), and (37), the convolution that yields thestamp and substrate deflections re-scales the trial pressure−compliancedistribution, p_(g)[m, n], to a real instantaneous pressure distributionby using the factor p₀(t_(h))/p_(g,0,u). This re-scaling is necessarybecause the responses of the stamp and substrate are instantaneous andnot time-dependent, and their kernels are generated using their trueelastic moduli.Compute w_(die,u)[m, n] and w_(local,u)[m, n]

Updated local (feature-level) polymer layer deformations are calculatedas follows within the contact region C:

w _(local,u) [m,n]=r ₀ −r _(inst,u-1) [m,n]−w _(die,u-1) [m,n]−k_(inst,u-1) [m,n]p _(g,u) [m,n]  (38)

and the updated polymer layer deformations associated with intra-dieinteractions are given by:

w _(die,u) [m,n]=w _(die,u-1) [m,n]+k _(die,u-1) [p _(g,u) [m,n]−p_(g,u-1) [m,n]]*g _(die,u-1) [m,n].  (39)

Compute r_(u)[m, n] According to p_(g,u)[m,n]

The updated residual layer expression r_(u)[m, n] is found as anincremental change from r_(u-1)[m, n]:

r _(u) [m,n]=r _(u-1) [m,n]+k _(inst,u-1) [m,n](p _(g,u) [m,n]−p_(g,u-1) [m,n])−└k _(die,u-1) [m,n](p _(g,u) [m,n]−p _(g,u-1) [m,n])┘*g_(die,u-1) [m,n]  (40)

Compute V/V₀[m, n]

In any regions of the stamp that are defined with finely patternedfeatures, the filled proportion of their volume may be estimatedaccording to the conservation of resist volume, using the followingrelation:

$\begin{matrix}{{\frac{V}{V_{0}}\left\lbrack {m,n} \right\rbrack} = \frac{{w_{local}\left\lbrack {m,n} \right\rbrack}{h\left\lbrack {m,n} \right\rbrack}}{1 - {\rho \left\lbrack {m,n} \right\rbrack}}} & (41)\end{matrix}$

where h[m, n] is the cavity height, and p[m, n] is the areal density ofprotrusions from the stamp. In places where this solution impliesV/V₀[m, n]>1, we clip V/V₀[m, n] to equal 1 and clip w_(local,u)[m, n]correspondingly, to (1−p)/hCompute k_(die,u)[m, n]

The scale factors k_(die,u-1)[m, n] account approximately for the effectof the spatial variation of r. Where the spatial average of r is muchsmaller than d, values of k_(die,u-1)[m, n] are proportional to the cubeof r_(u-1)[m, n]; where r is much larger than d, k_(die,u-1)[m, n]≈1.Where r˜d, k_(die,u-1)[m, n] may be calculated using a ‘featurediameter’ assumed to be given by the discretization pitch d andk_(die,u-1)[m, n] and to be inversely proportional to dp_(g)/dr′.

This approach may lose validity if layer thickness variation is a largeproportion of the average thickness, and in cases where d<<r butmaterial is displaced laterally at the die-scale over distances that arecomparable with or much greater than r.

Update the Squeeze-/Bulk-Deformation Switch S_(u)[m, n]

It is now necessary to update the estimate of which regions of the layerare operating in a squeeze-dominated mode and which arebulk-deformation-dominated. Firstly, the calculated pressure−compliancedistribution p_(g)[m, n] is used to determine residual layer-thicknessvalues r_(squeeze)[m, n] that correspond to squeeze-dominateddeformation of the layer (i.e. no cavity filling is considered).Secondly, p_(g)[m, n] is substituted into

$\begin{matrix}{\frac{V}{V_{0}} = \left\{ \begin{matrix}\frac{p_{g,0}{{sA}/h}}{V_{T\; 0}} & {{p_{g,0}{{sA}/h}} < 1} \\{1 - {\left( {1 - V_{T\; 0}} \right){\exp \left\lbrack \frac{1 - {p_{g,0}{{sA}/h}}}{k_{f}} \right\rbrack}}} & {{p_{g,0}{{sA}/h}} > 1.}\end{matrix} \right.} & (42)\end{matrix}$

The resulting values of V/V₀[m, n] are scaled to give absolute stampdisplacements, which are subtracted from r₀ to give a set of residuallayer-thickness values r_(bulk)[m, n] corresponding to bulk deformationof the layer. In calculating r_(bulk)[m, n], squeezing-limited flow ofthe residual layer is ignored.

Where r_(squeeze)[m, n]>r_(bulk)[m, n], it is deduced that resistance tosqueezing flow of the residual layer is limiting the ability of thestamp to penetrate the layer, and S_(u)[m, n] is set to 1. Otherwise, itis concluded that movement of the stamp into the layer is bulkdeformation-dominated (i.e., limited by forces distributed throughoutthe whole layer) and S_(u)[m, n] is set to zero. In this calculation,any lateral displacement of material, w_(die)[m, n], is ignored; theestimate of S_(u)[m, n] is based solely on the currentpressure−compliance distribution.

Update the Contact Set C

At the end of each step, the contact region C is also revised, subjectto the constraint that regions [m, n] with a negative pressure in thecurrent solution must be removed from C, while those regions in whichthe simulated polymer topography overlaps the stamp topography are addedto C.

A Strategy for Choosing Step-Size

In order to minimize the number of steps U necessary to complete asimulation, the following approach could be adopted. An initial stepsize for p_(g,0) is chosen that is typically around 5 or 10% ofp_(g,0,U). The solution for that step is performed. If any value of r[m,n] is negative in this new solution, this is taken to indicate that thestep-size was too large to represent reality, and the step solution isre-done using a step-size half as large as previously. If, on the otherhand, no value of r[m, n] is negative in the new solution, the algorithmadvances to the next step by incrementing p_(g,0) twice as much as forthe previous step. In this way, the algorithm should gravitate towardsthe largest step-size that allows the solution to proceed in a stablefashion.

The fidelity of simulation results reduces as the step-size isincreased: the pressure−displacement relations of the polymeric layerare linearized at each step, and the larger the step, the further alinearized solution will deviate from reality. Care therefore needs tobe taken to restrict the maximum allowable step-size for p_(g,0).

Demonstration of Die-Scale Simulation in ‘Flat’ Mode

As a demonstration of the simulation method, the experimental resultsfrom the imprinting of a simple test pattern with the results of asimulation may be compared while assuming a Newtonian resist model. Thesimulation is operated here in ‘flat’ mode, whereby the design of thestamp is discretized at the sub-feature scale and there is no finetopography included in the stamp's description.

Test Pattern

The test pattern includes features that are 5 to 50 μm in diameter andthe pattern is arrayed on a square grid with pitch 425 μm. The polarityof the pattern is arranged such that the stamp now contains a series ofsquare and rectangular protrusions.

Experimental Method

The test pattern was transferred to 1 μm of OCG825 photoresist on a 6″silicon wafer. The silicon was etched to a depth of ˜1 μm using SF₆chemistry and the resist was stripped using oxygen plasma.

The layer to be imprinted was prepared as follows. 950 kg/mol PMMA,dissolved 3% by weight in anisole, was spun on to a fragment of siliconthat was ˜10 mm square. The spinning conditions were 2000 rpm for 1minute. The prepared substrate was baked in an oven at 170° C. for 30minutes.

The PMMA layer on silicon was then placed into contact with part of thearray of test patterns on the silicon stamp. A piece of Neoprene rubber,˜8 mm square and 1 mm thick, was placed on the back of the 10 mm squaresilicon sample. The stamp/substrate/Neoprene stack was placed betweenthe platens of the hot embossing apparatus described in Chapter 2. Theplatens were heated to 165±1° C. and a sample-average pressure of 40 MPawas applied for two minutes. The platens were then cooled toapproximately 90° C. and the load was removed.

Metrology

A few small scratches were made through the imprinted resist layer toenable the position of the PMMA/silicon interface to be determined.Several regions of the sample were photographed in an optical microscopeand the interferogram obtained from one particular copy of the testpattern near the center of the sample.

FIG. 13 a-13 c illustrate optical micrographs and white-lightinterferometry cross-sections of test patterns. In FIG. 13 aexperimental results obtained from one particular copy of the testpattern near the center of the sample are shown. The color density ofeach region of the interferogram can be used to infer the approximatethickness of the imprinted layer.

To gain more precise measurements of the imprinted topography, awhite-light interferometer (Zygo NewView) was used. The PMMA layer istransparent and its thickness is too small to use white-lightinterferometry directly on the sample: reflections from the PMMA/Siinterface would have made it impossible for the interferometer'ssoftware to determine the true position of the PMMA/air interface.Depositing a reflective layer (e.g., sputtered gold) on the sample tostop light from penetrating the PMMA layer was also not feasible: therequired thickness of the deposited layer would have been comparable tothe relief of the imprinted topography. Instead, Sylgard 184 PDMS wascast on to the imprinted sample, cured at 80° C. for six hours, and thenpeeled from the sample. This PDMS casting was sufficiently thick thatwhite-light interferometry could be used directly on the surface of thecasting without encountering parasitic reflections from the flatback-side of the PDMS layer. The topographies obtained were then simplyflipped to give a representation of the imprinted PMMA topography.

It is known that white-light interferometry can produce erroneous datanear step-changes in topography, and indeed the data obtained from thissample do exhibit sharp spikes near feature edges. We cannot be surewhether these spikes are real or artifacts of the measurement procedure,and so the data were smoothed before being plotted in FIG. 13 a.

FIG. 14 is a plot that illustrates determination of initial PMMA layerthickness using scanning white-light interferometry results. The spun-onPMMA layer thickness was determined to be approximately 260 nm bylooking at the step-height of PMMA at the edge of one of the scratchesintroduced to the PMMA samples. This measurement was performed in aregion of the sample that had not made contact with any part of thestamp.

Newtonian Simulation

The simulation method described above was used to simulate the embossingof this PMMA sample. A Newtonian resist model was chosen with aviscosity of 5×10⁷ Pa·s, which is a plausible value for PMMA at 165° C.according to the data reported by Han [3]. The stamp and substrate,which are both silicon, are modeled with Young's modulus 160 GPa andPoisson's ratio 0.27. The nominal number of simulation steps, U, was setto 10.

The simulation results are plotted in FIG. 13 b. The simulated opticalinterferogram was constructed assuming a refractive index of 1.49 forPMMA. The simulation, which was discretized at a lateral pitch of 1 μm,took approximately six minutes to run.

Substantial stamp/substrate deformation is predicted by the simulation,leading to variation of the simulated depth of imprinting by more than afactor of two across the pattern. The general shape of these stampdeflections, whereby the larger features are imprinted to a muchshallower depth than the narrower ones, roughly follows that of theexperimental data. Tracking the simulated contact pressure distributionover time, it can be seen that the pressure is initially higher in theregion of the small, sparse stamp protrusions near the lower left cornerof the pattern, but that as the resist is progressively imprinted inthese regions the bulk of the contact pressure transfers to the widerfeatures on the upper and right-hand sides of the pattern.

It is important to note that this topographical nonuniformity ispredicted in the absence of any ‘springback’ of the resist materialitself: the nonuniformity is entirely attributable to deflections of thestamp and substrate.

Since it would be difficult to increase the stiffness of the stamp muchabove that of silicon, it is apparent that, for a pattern of thisnature, an improvement of residual layer uniformity would require eithera substantial extension of imprinting time, or a substantial reductionof resist viscosity enabling imprinting to be carried out at much lowerpressures.

The prediction of this Newtonian resist model is far from perfect,however: the exact measured topography of resist within each of thelarger features is not well captured. The experimental data in fact showmuch sharper gradients of topography near the edges of these largerfeatures, indicative of shear-thinning behavior.

Modeling Shear-Thinning

In order to represent shear-thinning in simulation, certain embodimentsdeal only with viscous (not viscoelastic) resist materials. The idea isto perform the simulation in a series of explicit time-steps, with eachstep u made up of two components. In the first component, the contactpressure distribution is estimated using a linear viscous model for theresist and taking as one of its inputs the layer topography r_(u-1)[m,n]from the previous step, u−1. The following equations are defined interms of real pressures p[m, n]:

p _(u) ^(f) [m,n]*g _(ss) [m,n]+{k _(die,u-1) [m,n]p _(u) ^(f) [m,n]}*g_(die,u-1) [m,n]Δt=−w _(stamp) [m,n]−r ₀ +r _(u-1) [m,n]  (43)

and

p _(u) ^(e) [m,n]*g _(ss) [m,n]±{k _(die,u-1) [m,n]p _(u) ^(e) [m,n]}*g_(die,u-1) [m,n]=1.  (44)

Δt is the length of step u, and k_(die) and g_(ss) are as defined above.Here, g_(die,u) is generated by substituting a ‘Young's modulus’ of 3 η₀into the function defined by Nogi and Kato, where η₀ is the zero-shearviscosity of the layer. The layer thickness substituted is the spatialaverage of the layer thickness r_(u-1)[m, n] from the previous step. Wehave the additional constraint that contact pressure is zero outside C:

p_(u)[m,n]=0∀m,n∉C.  (45)

Pressure distributions p^(f) _(u)[m, n] and p^(e) _(u)[m, n] are foundusing the stabilized biconjugate gradient algorithm. Having found thesetwo distributions, we solve for the incremental stamp referencedisplacement Δ_(u) such that:

p _(u) ^(e) [m,n]+Δ _(u) p _(u) ^(f) [m,n]=p _(u) [m,n]  (46)

subject to the constraint that:

$\begin{matrix}{{\sum\limits_{m = 1}^{M}{\sum\limits_{n = 1}^{N}{p_{u}\left\lbrack {m,n} \right\rbrack}}} = {{MNp}_{0,u}.}} & (47)\end{matrix}$

An amount of linear viscous flow consistent with this pressuredistribution p_(u)[m, n] and with the length Δt of the time-step is thencomputed:

Δr _(lin,u) [m,n]=−(p _(u) [m,n]k _(die,u-1) [m,n])*g _(die,u)[m,n]Δt.  (48)

In the second component of each step, an additional amount of plasticflow associated with shear-thinning is computed. We expect the rate ofadditional flow at a given location in the resist to be related to themagnitude of in-plane shear stress at that location. The magnitude ofin-plane shear stress can readily be calculated from thecontact-pressure distribution p_(u)[m, n] by way of two additionalkernel functions defined by Nogi and Kato. Where the plane of the resistis x−y, the Fourier transforms of the in-plane shear stresses in thelayer arising in response to unit pressure applied at location [0, 0]are given by:

$\begin{matrix}{\left. \mspace{79mu} \begin{matrix}{{S_{yz}\left\lbrack {m,n} \right\rbrack} = {- {\sqrt{- 1}\left\lbrack {{\frac{m\; \pi}{Nd}{\alpha \left( {A - \overset{\sim}{A}} \right)}} + {z\frac{n\; \pi}{Nd}{\alpha \left( {B - \overset{\sim}{B}} \right)}}} \right\rbrack}}} \\{{S_{zx}\left\lbrack {m,n} \right\rbrack} = {- {\sqrt{- 1}\left\lbrack {{\frac{n\; \pi}{Nd}{\alpha \left( {A - \overset{\sim}{A}} \right)}} + {z\frac{m\; \pi}{Nd}{\alpha \left( {B - \overset{\sim}{B}} \right)}}} \right\rbrack}}}\end{matrix} \right\} \mspace{85mu} {where}} & (49) \\{\left. \begin{matrix}{{\alpha = \sqrt{\left( \frac{m\; \pi}{Nd} \right)^{2} + \left( \frac{n\; \pi}{Nd} \right)^{2}}};{\lambda = {- 1}};{\kappa = {- 1}}} \\{A = {\left\lfloor {0.5\left( {\kappa - \lambda - {4{\kappa\alpha}^{2}r_{mean}^{2}}} \right){\exp \left( {{- 2}\; \alpha \; r_{mean}} \right)}} \right\rfloor R\; {\exp \left( {{- \alpha}\; z} \right)}}} \\{\overset{\sim}{A} = {\left\lfloor {0.5\left( {\kappa - \lambda - {4{\kappa\alpha}^{2}r_{mean}^{2}}} \right){\exp \left( {{- 2}\; \alpha \; r_{mean}} \right)}} \right\rfloor R\; {\exp \left( {{- \alpha}\; z} \right)}}} \\{B = {\left\lbrack {1 - {\left( {1 - {2\alpha \; r_{mean}}} \right)\kappa \; {\exp \left( {{- 2}\alpha \; r_{mean}} \right)}}} \right\rbrack \alpha \; R\; {\exp \left( {{- \alpha}\; z} \right)}}} \\{\overset{\sim}{B} = {\left\lbrack {1 + {2\alpha \; r_{mean}} - {\lambda \; {\exp \left( {{- 2}\alpha \; r_{mean}} \right)}}} \right\rbrack \kappa \; {\exp \left( {{- 2}\alpha \; r_{mean}} \right)}\alpha \; R\; {\exp \left( {{- \alpha}\; z} \right)}}}\end{matrix} \right\} {R = {{- \left\lfloor {1 - {\left( {\lambda + \kappa + {4\kappa \; \alpha^{2}r_{mean}^{2}}} \right){\exp \left( {{- 2}\alpha \; r_{mean}} \right)}} + {\lambda \; \kappa \; {\exp \left( {{- 4}\alpha \; r_{mean}} \right)}}} \right\rfloor^{- 1}}{\alpha^{- 2}.}}}} & (50)\end{matrix}$

The thickness defined for the layer, r_(mean), is the spatial average ofr_(u-1)[m, n]. The functions above are defined for an incompressiblelayer and a perfectly stiff substrate (substrate compliance is allowedfor elsewhere, in g_(ss)[m, n]). The quantity z is the distance betweenthe top surface of the layer and the plane in which shear stresses areexpressed; shear stress at a particular location [m, n] is proportionalto z, provided that z≦r_(mean). S_(yz)[m, n] and S_(zx)[m, n] areevaluated over a range of 2N in both m and n, taking z=0.5r so that theshear stresses computed are those half-way through the thickness of thelayer. The values of S_(yz)[0,0] and S_(zx)[0,0] are undefined accordingto equation (49); however, the average value of shear stress in thelayer in response to a normally applied load is zero, so we set bothS_(yz)[0,0] and S_(zx)[0,0] to zero. After inverse Fouriertransformation of S_(yz)[m, n] and S_(zx)[m, n], the central N×N regionsof the resulting matrices are taken as the kernel functions, s_(yz)[m,n] and s_(zx)[m, n], for use in simulation.

The in-plane shear stress components are then given by convolution:

τ_(yz) [m,n]=s _(yz) [m,n]* p _(u) [m,n]; τ _(zx) [m,n]=s _(zx) [m,n]*p_(u) [m,n].  (51)

We define the magnitude of in-plane shear stress as follows:

τ _(u) [m,n]=√{square root over (τ_(yz) ² [m,n]+τ _(zx) ² [m,n])}.  (52)

For any given temperature there is a breakpoint in strain-rate abovewhich a Newtonian model ceases to hold and viscosity falls withincreasing strain rate. From the perspective of shear stress, thisbehavior can be described in terms of the existence of a yield stress,below which the rate of resist flow is proportional to stress and abovewhich it increases super-linearly with stress. Since we are trying toexpress an additional incremental amount of plastic flowΔr_(thinning,u)[m, n] beyond that implied by a Newtonian model, wepropose the following expression:

$\begin{matrix}{{\Delta \; {r_{{thinning},u}\left\lbrack {m,n} \right\rbrack}} = {{- \left( {{p_{u}\left\lbrack {m,n} \right\rbrack}{k_{{die},{u - 1}}\left\lbrack {m,n} \right\rbrack}} \right)}{\max \left\lbrack {{{{\overset{\_}{\tau}}_{u}\left\lbrack {m,n} \right\rbrack} - \tau_{yield}},0} \right\rbrack}^{b}{\frac{k_{shear}\Delta \; t}{\eta_{0}}.}}} & (53)\end{matrix}$

where τ_(yield) is a yield shear stress for the layer, k_(shear) is aconstant depending on the thickness of the layer and the pitch d ofdiscretization, and b is an exponent describing the shear strain rate'sdependence on shear stress.

The layer topography r_(u)[m, n] that is carried forward to the nextstep is then given by:

r _(u) [m,n]=r _(u-1) [m,n]+Δr _(lin,u) [m,n]+Δr _(thinning,u)[m,n]  (54)

The overall idea of this approach is that by calculating the evolvingpressure distribution with a linear model, we can still use theconvolution approach of the previous sections, and the speed of solutionwill be much greater than if the non-linear shear-thinning model were tobe integrated in the pressure-solution step. The value of r_(u)[m, n] isupdated separately at the end of each step, in order to keep track ofthe additional, shear-thinning-related flow.

In trying to implement this approach, it was found that after the firstcomplete solution step the calculated pressure distributions started tobecome very rough indeed. Further work is needed to resolve thesenumerical problems. An illustration of the scheme, however, can beprovided by looking at the output of the model after just one step,where that step has been specified to have a length equal to the entireduration of the imprinting cycle. Such an illustration is given in FIG.13( c). The simulation was performed at a lateral discretization, d, of1 μm. The value of b found to represent experimental results effectivelywas 0.4, that of τ_(yield) was 0.45 MPa, and that of k_(yield) was5.1×10⁻⁵ m/(Pa^(0.4)). The duration of the loading cycle was 120 s, thepattern-average applied pressure was 40 MPa, and the zero-shearviscosity specified was 5×10⁷ Pa·s. Elastic properties of the stamp andsubstrate were as described above. The simulation results shown in FIG.13( c) bear a substantially closer resemblance to the experimentallyobtained topography (FIG. 13( a)) than does the simulation that assumesa Newtonian resist (FIG. 13( b)). The shear-thinning simulationsuccessfully represents the bands of ˜10 μm width around the edges ofthe larger features have been imprinted deeply into the resist layer.These bands correspond to the locations of in-plane shear stresses thatexceeded the fitted yield stress of 0.45 MPa.

Because only one simulation step was used here, we cannot expect theaccuracy of the simulated topography—or indeed the validity of theextracted values of b, τ_(yield) and k_(yield)—to be particularly high.Even so, the proposed method of using in-plane shear stresses to computethe locations of shear-thinning behavior does appear promising. Thetotal simulation time for this crude, single-step solution was only twominutes.

The PMMA sample analyzed here was processed under conditions where agreat amount of shear thinning evidently occurred. In many realisticimprint cases, though, the resist would be processed much further aboveT_(g), so that its behavior would in general be much closer toNewtonian. Therefore, we expect the incorporation of shear-thinningcomputations to be required in some, but by no means all, nanoimprintsimulations.

Computing Stamp Deflections when Cavities Fill Quickly with Resist

In some imprint cases, the viscosity of the resist upon application ofthe stamp is sufficiently low that all stamp cavities are readily filledduring the loading cycle. In these cases, it may be safe to begin withthe assumption that every part of the stamp is in contact with resistmaterial and to proceed from there with a solution for the deflectionsof the stamp and substrate. When this assumption is possible, we canavoid the simulation time that would have been taken to estimate thesize and shape of the contact region.

In cases where the cavities of the stamp are completely filled withpolymer, and assuming that the resist material is a viscousincompressible fluid (which is to be expected if the resist is able tofill the stamp quickly), the only net motion of stamp towards substratewill be that associated with global squeezing of resist towards theedges of the wafer. Within a small, die-sized region of the stamp orwafer, we could characterize the velocity of the stamp as beinginfinitesimally small and uniform across the region.

Our current simulation methods implicitly assume any modeled pattern tobe both periodic and infinite in lateral extent, and it is impossiblewithin this framework for there to be a non-zero steady-state uniformvelocity of the stamp when all cavities are filled and the resist isincompressible: net lateral transport of material into or out of thesimulation region is inconsistent with its boundary conditions. Yet weknow this to be an imperfect representation of a wafer being imprinted:some small amount of material can be squeezed laterally from die to die,towards the edge of the wafer. We therefore make the approximation ofneglecting the very small global squeezing motion of the stamp andfinding a quasi-steady-state pressure distribution that gives zero stampvelocity at every location within the simulation region.

In the ‘flat’ simulation approach described in the previous section, thecontact pressure scaling factor k_(die) was introduced to capture theidea that, for a particular pressure applied to any given region of aresist layer, the region squeezes down more slowly the thinner it is.Where the lateral dimensions of the imprinted pattern are substantiallylarger than the layer thicknesses, it was proposed that k_(die) shouldbe proportional to the cube of local thicknesses, reflecting the knownproperties of squeeze-film flow. The concept of k_(die) is also appliedhere in the filled-stamp case, the reason being that just beforequasi-steady-state was achieved, there was the possibility of motion ofthe stamp and these squeezing rules would have applied. The followingrelation is considered to hold for step u of an iterative solutionprocess:

(k _(die,u-1) [m,n]p _(u) [m,n])*g _(die,u-1) [m,n]=0  (55)

If the layer were uniform in thickness, k_(die) would be 1 everywhereand p_(u) would be uniform, whatever the shape and amplitude of g_(die).Based on equation (55) alone, the magnitudes of p_(u)[m, n] areindeterminate, but their spatial average is usually specified for anyimprinting process.

If we have an estimate of the current residual layer thicknessr_(u-1)[m, n], we can directly estimate k_(die,u-1)[m, n], and hencep_(u)[m, n], which is expected to be inversely proportional tok_(die,u-1)[m, n] and is subject to the constraint that its spatialaverage must equal the applied imprinting pressure. The deflections ofthe stamp and substrate can then be straightforwardly computed byconvolving p_(u)[m, n] with the elastic point-load response function ofsilicon. Based on stamp and substrate deflections, revised estimates ofr_(u)[m, n] and hence k_(die,u)[m, n] can be made. The more iterationsperformed, the more accurate the simulation is expected to be.

The initial values of the resist layer thicknesses r₁[m, n] areestimated by assuming the stamp to be un-deformed but embedded in theresist such that the resist's initial volume is displaced to fill allstamp cavities.

It is important to note that is it not necessary to know the absoluteviscosity of the resist to employ this method: merely the topography andelasticity of the stamp and substrate, the initial resist layerthickness, and the spatial average of p[m, n].

This approach was used in an attempt to simulate the reportedexperimental results of Pedersen et al., who used a silicon stamp toimprint simple patterns into mr-I 7030 resist [1]. Imprinting wasperformed at 140° C. under 0.65 MPa for 5 minutes. The relief of thestamp was 300 nm and the apparent initial polymer layer thicknessesapproximately 360 nm. Pedersen et al. measured the residual layerthickness variation across the resist patterns after removal of thestamp.

The reported imprinting temperature is well above the literature valueof 60° C. for this particular resist's glass-transition temperature.There is no evidence in the reported results of elastic springback ofthe residual layer (even though the authors refer to it), so it seemsappropriate to adopt a purely viscous model for this simulation. Thezero-shear viscosity of mr-I 7000-series resists at 140° C. has beenreported by its manufacturers to be 4×10³ Pascal [3]. This viscosity isabout four orders of magnitude lower than that of the PMMA processed at165° C. in the previous section, so it is reasonable that stamp cavitieswould be expected to fill comparatively quickly with this mr-I resist.

The lateral extent of the patterns shown in Pedersen's work is ˜1.5 mm,and we assume that there were no stamp protrusions in the regionsurrounding them, so we choose to model a region that is 3 mm square andotherwise empty, with the expectation that the pattern periodicityimplicit in our approach will then not substantially perturb results.For the silicon stamp and substrate, a Young's modulus of 130 GPa and aPoisson's ratio of 0.27 are assumed.

FIG. 15 a-15 c illustrate results of the cavity filled simulation. Theshape and magnitude of stamp deformation have been modeled quitesuccessfully using the known properties of the silicon stamp andsubstrate and the known applied average pressure.

In attempting to represent the imprinting of a modified grating withdummy features at the sides, the approach as described so far in thissection is less successful. In FIG. 15 a (Approach 1), the results ofthe basic simulation method are shown: they do not fully capture theobserved center-high ‘bulge’ of residual layer thickness of Pedersen etal. In this experimental case, resist material in the central five stampcavities is laterally enclosed on all four sides by stamp protrusions.It might therefore be appropriate to regard this material as being‘unavailable’ for lateral flow. In that case, k-_(die) in these regionsshould be set to the value consistent with a residual layer thicknessthat does not include this unavailable material.

FIG. 16 illustrates the concept of a region of laterally enclosed resistin a filled cavity or cavities. The effect of this modeling change is toincrease substantially the simulated contact pressure in the regionscontaining resist material that is ‘unavailable’ for flow. At eachiteration step, any material ‘unavailable’ for flow does of coursecontinue to contribute to the total volume of resist under the stamp.

Making this change manually to the simulation approach, a stampdeflection profile is obtained that is much closer to that measuredexperimentally: FIG. 15 b (Approach 2). Ten iteration steps u were foundto be ample to obtain a stable solution, and each of the simulationsshown was computed within ˜10 s using the equipment described earlier inthis thesis. Because the contact set is full, there is no need to solveiteratively for a pressure distribution at each step u: the simulationsimply consists of a series of forward convolutions followed by directre-computation of k_(die) and r.

While no firm conclusions can be drawn by studying this singleexperimental sample, the example presented here suggests three things.Firstly, it is possible that when a resist material readily fills stampcavities, stamp deflections and hence residual layer nonuniformity canbe computed much more quickly than in cases where the contact region hasto be predicted through simulation. Secondly, although Pedersen et al.interpret ‘springback’ of the resist layer itself as a cause of theresidual layer thickness nonuniformity, our model indicates that stampand substrate deflection alone can explain the profiles observed.Thirdly, in cases where the stamp cavities are completely filled, it maybe necessary to distinguish between cavities that are completelylaterally enclosed and those that are open on one or more sides.

This third observation, if correct, potentially creates the need foradditional algorithms to detect fully laterally enclosed cavities inarbitrary patterns. Further experimentation will clearly be needed toestablish how best to model filled-stamp cases.

A Refined Approach to Die-Scale NIL Simulation

The above analysis of Pedersen's experimental results indicated thatwhen a layer of material being imprinted varies laterally in thickness,the pressure at a particular point in the layer depends not only on thelocal thickness but also on the extent to which that material islaterally constrained. For example, a particularly thick portion of alayer fully surrounded by much thinner portions will be rather harder toreduce in thickness by squeezing than will a region of that particularthickness surrounded by thicker regions.

As explained above, in a die-scale simulation approach the quantityk_(die)[m, n], linking local contact pressure to the resultingdisplacement of material away from point [m, n], is defined as afunction of the instantaneous layer thickness at point [m, n] alone. Forlayers much thinner than the lateral pitch of the simulation grid,k_(die)[m, n] was proposed to be proportional to the cube of the locallayer thickness. If residual layer thickness changes very gradually withlateral position, this existing approach can be satisfactory, but incases where stamp cavities and hence layer thicknesses exhibitstep-changes in height (as in almost any feature-scale simulation) theexisting approach implies step changes in local pressure and is notlikely to produce realistic results. What is needed instead is a methodthat accounts for the degree of lateral confinement of each portion ofan imprinted layer.

A simple way of accounting for the lateral confinement of material issuggested here. We present the idea in two-dimensional form although itis expected that the method could be readily extended to threedimensions. FIGS. 17 a-17 b illustrate a generalization of thesimulation of thin-layer squeezing for varying layer thicknesses andlateral constraints (1-D case). For comparison, FIG. 17 a shows themethod that has been employed thus far: the mechanical impulse responsefunction of a viscous imprinted layer effectively takes the form of atriplet, provided that the pitch of lateral discretization is muchlarger than the layer thickness. For the squeezing of an infinitely longstrip of material between a rigid stamp protrusion and a rigidsubstrate, a parabolic pressure distribution is consistent with thiskernel function and with theory. In FIG. 17 b, in contrast, we considerthe case in which the same layer material is in contact with a rigidstamp protrusion having half the width of that in FIG. 17 a. Theleft-hand side of the layer material is prevented from moving leftwardsby a ‘sluice’, attached to the stamp that is imagined to mate with thesubstrate, forming a perfect seal. By symmetry, therefore, we expect thepressure distribution in this layer to be identical to that in one halfof the layer illustrated in FIG. 17 a.

We now decompose the impulse response of the layer into two parts: one,g_(A)[n], representing the displacement of material rightwards from thepoint of application of a load, and another, g_(B)[n], representingleftwards displacement. Note that the sum of these sub-kernels equalsthe triplet kernel in FIG. 17 a. We also define two so-called ‘masking’functions, k_(A)[n] and k_(B)[n], which describe any lateral constraintson rightwards and leftwards material displacement respectively. In thissimple case, there are no constraints on rightwards materialdisplacement, and k_(A)[n] equals 1 everywhere, but there is aconstraint preventing leftwards displacement of material from theleftmost region of the imprinted layer, and k_(B) consequently equalszero in that region and one elsewhere. We then find the pressuredistribution p[n] that is the solution to the following set of linearequations:

$\begin{matrix}{\frac{{r\lbrack n\rbrack}}{t} = {{{g_{A}\lbrack n\rbrack}*\left( {{k_{A}\lbrack n\rbrack}{p\lbrack n\rbrack}} \right)} + {{g_{B}\lbrack n\rbrack}*\left( {{k_{B}\lbrack n\rbrack}{p\lbrack n\rbrack}} \right)}}} & (56)\end{matrix}$

where r[n] is the topography of the upper surface of the imprintedlayer, and the material is assumed to exhibit Newtonian viscosity.Additionally, p[n] is constrained to be zero outside the region ofstamp−layer contact. As shown in FIGS. 17 a-17 b, the pressuredistribution found with this method is the half-parabola expected.

In cases where there are step changes in layer thickness rather than thecomplete blockage of lateral material flow, k_(A)[n] and k_(B)[n] couldtake values other than zero or one. We suggest the following rule forsetting their values based on instantaneous layer thicknesses r[n]:

$\begin{matrix}{{{k_{A}\lbrack n\rbrack} = \frac{\min \left( {{r\lbrack n\rbrack}^{3},{r\left\lbrack {n + 1} \right\rbrack}^{3}} \right)}{r_{0}^{3}}}{{k_{B}\lbrack n\rbrack} = \frac{\min \left( {{r\lbrack n\rbrack}^{3},{r\left\lbrack {n - 1} \right\rbrack}^{3}} \right)}{r_{0}^{3}}}} & (57)\end{matrix}$

where r₀ is the average layer thickness across the simulation region andthe amplitudes of g_(A)[n] and g_(B)[n] are computed for a layer ofthickness r₀. FIGS. 18 a-18 e includes examples of uses of thedecomposed-kernel approach to simulate several one-dimensionalgeometries.

The approach to calculating k_(A)[n] and k_(B)[n] was used to producethe pressure simulations that are illustrated in FIG. 18 a-18 c. FIGS.18 a-18 b reiterate the cases illustrated in FIGS. 17 a-17 b; meanwhileFIG. 18 c shows a case in which the ‘sluice’ at the left-hand side ofthe contact region has a finite opening height and is able to transmitsome material. In FIG. 18 d a split-level stamp is illustrated, wheresimulations have been performed for a range of height ratios. FIG. 18 eillustrates a case in which a stamp protrusion has had a taller cavityintroduced at its center, and we see that as the height of the cavityincreases, the pressure within that cavity approaches a uniformdistribution.

This simulation method may be performed iteratively, with small steps inlayer thickness being computed along with evolution of the stamp−layercontact set. Validation of this method may be provided by comparing itspredictions to those of full solutions of the Navier-Stokes equationsfor some test geometries. We anticipate that this method may be extendedto three-dimensional simulation by employing four or eight sub-kernels,one corresponding to each topographical cell neighboring any givenlocation.

In cases where the pitch of lateral discretization is comparable to orless than the layer thicknesses, the kernel is not able to be simplifiedto a triplet as it was in FIG. 17 a. It may however still be possible touse a modified form of this approach and to decompose an appropriatenon-triplet kernel function into several directional sub-kernels.

The simulation approach is fast to run and the resolution with whichresults are delivered can be arbitrarily specified. A simulation using a64×64 matrix-representation of a pattern has been successfully completedin less than a minute. Outputs of the model are residual layerthickness, contact pressure distribution, and the proportions of cavityvolumes filled.

The use of a particularly soft stamp material has the effect ofproducing imprinted ‘grooves’, located at the edges of stamp featuresand with a width comparable to that of the imprinted layer. This effectcan be exploited to create small features in resist from much largerones on a stamp.

In nanoimprint lithography, what is needed from simulation is not simplya prediction of whether stamp cavities will fill with polymer, but alsoan impression of the uniformity of thickness of the polymer's residuallayer. Certain embodiments of the present invention simulate nanoimprintand provide both of these required model outputs. Although thedescription herein focuses on the kind of nanoimprint in which a uniformlayer of thermoplastic polymeric resist is spun on to a hard substrateand then imprinted, various nanoimprint problems may be approached usingaspects of the present invention.

FIG. 19 demonstrates an example embodiment 1900 of the present invention1900 for iterative fitting of material parameters of deformable body.The example embodiment 1900 employs embossing stamp design 1910,experimental parameters 1920 (such as temperatures, pressures, loadingdurations used in experiments), and candidate material model parameters1930 (such as E(T), η(T), shape of point load-time response) to obtainsimulated deformed topography (or topographies) of a deformed body 1950.The example embodiment 1900 employs forward simulation for each set ofexperimental processing parameters (shown later in FIG. 20) 1940.

The example embodiment 1900 determines if the simulated topographiesmatch measured topographies closely 1960. If so 1965, the exampleembodiment 1900 reports the obtained material model parameters 1970. Ifthe example embodiment 1900 determines that the simulated topographiesdo not match measured topographies closely 1968, the example embodiment1900 refines candidate material model parameters 1980 and forwards therefined model parameters 1990 for so that forward simulations can berepeated 1940. In determining whether or not the simulated topographiesmatch measured topographies closely 1960, the example embodiment 1900may consider other factors such as specification of fitting tolerance1955 (i.e., how closely does the simulation have to match the experimentto be satisfied), and experimentally measured topography (ortopographies).

FIG. 20 is a high level flow chart of the forward simulation proceduresaccording to an example embodiment 2000 of the present invention 2000.The example embodiment 2000 generates point load response function of avirtual elastic body 2006 based on the average pressure applied to backof stamp as a function of time 2002, temperature of deformable body as afunction of time 2004, and the material model of the deformable body aswell as the shape of point load-time response 2007. The exampleembodiment optionally (for thin deformable layers) may convolve thepoint load-time response with material spread function 2008.

The example embodiment 2000 employs the stamp design 2010 and sets aninitial guess for contact set C 2015. The example embodiment 2000employs the virtual elastic point load response 2009, the stamp design2012, and the contact set guess 2018 to iteratively find the stamp bodycontact pressure distribution that is consistent with contact set C,design of stamp (assumed rigid), and material virtual point loadresponse function. This involves convolution of point load response withpressure distribution 2020.

The example embodiment 2000 employs the simulated topography andpressure distribution 2030 to determine whether the simulated topographyof the deformable body implies intersection with stamp at one or morelocations 2035. If so 2038, the example embodiment 2000 addsintersecting elements to C 2040 and employs the revises contact setestimate C 2042 to determine if any of the simulated contact pressuresare negative 2044. If the simulated topography of the deformable bodydoes not imply intersection with stamp at one or more locations 2045,the example embodiment 2000 proceeds to determine if any of thesimulated contact pressures are negative 2044. If so 2046, the exampleembodiment 2000 removes from C elements having negative simulatedpressures 2048 and revises contact set estimate C 2050. If not 2045, theexample embodiment 2000 optionally may determine if the number ofiterations around the loop L 2080 has reached a predetermined number2060. If not 2068, the example embodiment 2068 repeats the loop L 2080using the revised contact set estimate C 2050. If the number ofiterations around the loop L 2080 has reached a predetermined number2060, the example embodiment reports topography and optionally pressuredistribution 2070.

FIG. 21 is a flow chart of an example embodiment 2100 of the presentinvention. The example embodiment 2100 employs Sample deformable bodies2110, experimental parameters (such as temperatures, pressures, loadingdurations to be used in experiments) 2120, and embossing test stamp 2130to deform deformable bodies and perform material characterizationexperiments, involving embossing with a test stamp 2140.

The example embodiment 2100 employs the deformed bodies 2145 to obtainmeasurement of deformed topographies (e.g. using white-lightinterferometry) 2150. The example embodiment 2100 employs the topographymeasurement data 2155 for iterative fitting of model (shown in FIG. 19)2160. The example embodiment 2100 employs the fitted material modelparameters 2165 such as E(T), η(T), and optionally shape of pointload-time response are in forward simulation of embossing (e.g. ofdevices to be manufactured) 2170 and verifies if simulated topographysatisfy specification for embossing process and/or embossed body 2180(specification for process (e.g. embosser capabilities, requirements forstamp cavity penetration by deformable body 2183 may also be consideredin this decision making process). If so 2182, the example embodiment2100 proceed to manufacturing using chosen parameters and stamp design.If not 2181, refines processing parameters p₀(t) and/or T(t), and/ordesign of stamp to be embossed 2175 and repeats the forward simulationof embossing 2170.

The example embodiment 2100 may also employs initial, unrefined,processing parameters p₀(t) and/or T(t), and unrefined design of stampto be embossed 2168 in the forward simulation of embossing 2170.

It should be understood that procedures, such as those illustrated byflow diagram or block diagram herein or otherwise described herein, maybe implemented in the form of hardware, firmware, or software, executedin any device such as a general purpose computer or an applicationspecific computer. If implemented in software, the software may beimplemented in any software language consistent with the teachingsherein and may be stored on any computer-readable medium known or laterdeveloped in the art. The software, typically, in form of instructions,can be coded and executed by a processor in a manner understood in theart.

An alternative embodiment of the claimed invention is presented in“Towards Nonoimprint Lithography-Aware Layout Design Checking,” byHayden Taylor and Duane Boning, (Proc. SPIE, Vol. 7641, 764129, Apr. 2,2010).

The teachings of all patents, published applications and referencescited herein are incorporated by reference in their entirety.

While this invention has been particularly shown and described withreferences to example embodiments thereof, it will be understood bythose skilled in the art that various changes in form and details may bemade therein without departing from the scope of the inventionencompassed by the appended claims.

References, all of which are incorporated by reference in theirentirety:

-   [1] R. H. Pedersen, L. H. Thamdrup, A. V. Larsen, and A. Kristensen,    “Quantitative Strategies to Handle Stamp Bending in NIL,” in Proc.    Nanoimprint and Nanoprint Technology, 2008.-   [2] T. Nogi and T. Kato, “Influence of a hard surface layer on the    limit of elastic contact—Part I: Analysis using a real surface    model,” Journal of Tribology, Transactions of the ASME, vol. 119,    1997, pp. 493-500.-   [3] J. K. Lee and C. D. Han, “Evolution of polymer blend morphology    during compounding in an internal mixer,” Polymer, vol. 40, November    1999, pp. 6277-6296.-   [4] H. Gao, H. Tan, W. Zhang, K. Morton, and S. Y. Chou, “Air    Cushion Press for Excellent Uniformity, High Yield, and Fast    Nanoimprint Across a 100 mm Field,” Nano Letters, vol. 6, September    2006, pp. 2438-2441.-   [5] K. Deguchi, N. Takeuchi, and A. Shimizu, “Evaluation of pressure    uniformity using a pressure-sensitive film and calculation of wafer    distortions caused by mold press in imprint lithography,” Japanese    Journal of Applied Physics, Part 1: Regular Papers and Short Notes    and Review Papers, vol. 41, 2002, pp. 4178-4181.-   [6] S. Timoshenko, Theory of Plates and Shells, New York:    McGraw-Hill, 1959.-   [7] J. N. Reddy, Theory and Analysis of Elastic Plates and Shells,    Boca Raton, Fla.: CRC, 2007.-   [8] D. Suh, S. Choi, and H. H. Lee, “Rigiflex lithography for    nanostructure transfer,” Advanced Materials, vol. 17, 2005, pp.    1554-1560.-   [9] G. McClelland, C. Rettner, M. Hart, K. Carter, M. Sanchez, M.    Best, and B. Terris, “Contact mechanics of a flexible imprinter for    photocured nanoimprint lithography,” Tribology Letters, vol. 19,    2005, pp. 59-63.-   [10] R. H. Pedersen, O. Hansen, and A. Kristensen, “A compact system    for large-area thermal nanoimprint lithography using smart stamps,”    Journal of Micromechanics and Microengineering, vol. 18, 2008,    055018.-   [11] H. Hocheng and W. H. Hsu, “Effect of Back Mold Grooves on    Improving Uniformity in Nanoimprint Lithography,” Japanese Journal    of Applied Physics, vol. 46, 2007, pp. 6370-6372.-   [12] T. Nielsen, A. Kristensen, and O. Hansen, “Flexible    Nano-Imprint Stamp,” U.S. patent application Ser. No. 11/574,645,    2005.-   [13] M. Colburn, A. Grot, B. J. Choi, M. Amistoso, T. Bailey, S. V.    Sreenivasan, J. G. Ekerdt, and C. G. Willson, “Patterning nonflat    substrates with a low pressure, room temperature, imprint    lithography process,” Journal of Vacuum Science and Technology B:    Microelectronics and Nanometer Structures, vol. 19, 2001, pp.    2162-2172.-   [14] H. K. Taylor and D. S. Boning, “Towards nanoimprint    lithography-aware layout design checking,” Proc. SPIE, vol. 7641,    764129, 2010.

1. A method of embossing with a stamp a deformable body comprising adeformable layer on a substrate, the method comprising: running adeformation model based on convolving a contact pressure distributionwith a mechanical response of a surface topography of the deformablelayer and convolving the contact pressure distribution with a point loadresponse of the stamp; and embossing the deformable body using the stampas a function of the deformation model.
 2. The method of claim 1 whereinthe deformation model is further based on convolving the contactpressure distribution with a point load response of the substrate. 3.The method of claim 1 further including determining the point loadresponse of the stamp based on bending of the stamp.
 4. The method ofclaim 3 further including determining the bending of the stamp based onthickness of the stamp.
 5. The method of claim 4 further includingdetermining the bending of the stamp based on elasticity of the stamp.6. The method of claim 3 further including determining the point loadresponse of the stamp based on indentation of the stamp.
 7. The methodof claim 1 wherein the deformable body includes a polymeric layercomparable in thickness to lateral dimensions of features beingembossed.
 8. The method of claim 1 wherein the deformable body includesa polymeric layer of thickness substantially less than lateraldimensions of features being embossed.
 9. The method of claim 1 furtherincluding determining the contact pressure distribution based on elasticdeflections of stamp and substrate.
 10. The method of claim 1 furtherincluding determining the contact pressure distribution based on stampindentation and substrate indentation.
 11. The method of claim 1 furtherincluding determining the contact pressure history distribution based ondeformations of a layer of the deformable body.
 12. The method of claim1 further including determining the contact pressure distribution basedon a zero-mean pressure distribution needed to bring all surfaces of thelayer of the deformable body into contact with the stamp.
 13. The methodof claim 1 further including determining the contact pressuredistribution based on a zero-mean pressure distribution needed for anincremental displacement of the stamp into the deformable body.
 14. Themethod of claim 1 further including determining the contact pressuredistribution based on a spatial variation of thickness of the layer ofthe deformable body.
 15. The method of claim 1 further includingdetermining the contact pressure distribution based on the point loadresponse and an average pressure applied to the stamp.
 16. The method ofclaim 1 further including determining the mechanical response of thesurface topography of the deformable body based on an average layerthickness of layers of the deformable body.
 17. The method of claim 1wherein the mechanical response of the surface topography of thedeformable body is anisotropic.
 18. The method of claim 1 wherein themechanical response of the surface topography is a spatial response. 19.The method of claim 18 wherein the mechanical response of the surfacetopography is a temporal response.
 20. A method of embossing with astamp a deformable body comprising a deformable layer on a substrate,the method comprising: running a deformation model determined based onconvolving a contact pressure distribution and a point load response,the point load response being an anisotropic function; and embossing thedeformable body using the stamp as a function of the deformation model.21. The method of claim 20 wherein the deformable body includes apolymeric layer thicker than dimensions of features being embossed. 22.The method of claim 21 further including determining a displacement ofmaterial in the deformable body embossed with the stamp as a function ofthe point-load-time response, one or more additional properties of thedeformable body, and the contact pressure distribution.
 23. The methodof claim 22 wherein the one or more additional properties of thedeformable body includes at least one of time dependent properties,temperature dependent properties, temperature and time dependentproperties, temperature dependent elasticity, or temperature dependentviscosity.
 24. The method of claim 20 further including determining theanisotropic point load response using an anisotropy parameter thatrepresents strength of anisotropy of the anisotropic layer of deformablebody.
 25. A system for embossing with a stamp a deformable bodycomprising a deformable layer on a substrate, the system comprising: aprocessor that runs a deformation model based on convolving a contactpressure distribution with a mechanical response of a surface topographyof the deformable layer and convolving the contact pressure distributionwith a point load response of the stamp; and an embosser that embossesthe deformable body using the stamp as a function of the deformationmodel.
 26. The system of claim 25 wherein the deformation model isfurther based on convolving the contact pressure distribution with apoint load response of the substrate.
 27. The system of claim 25 whereinthe processor determines the point load response of the stamp based onbending of the stamp.
 28. The system of claim 27 wherein the processordetermines the bending of the stamp based on thickness of the stamp. 29.The system of claim 28 wherein the processor determines the bending ofthe stamp based on elasticity of the stamp.
 30. The system of claim 27wherein the processor determines the point load response of the stampbased on indentation of the stamp.
 31. The system of claim 25 whereinthe deformable body includes a polymeric layer comparable in thicknessto lateral dimensions of features being embossed.
 32. The system ofclaim 25 wherein the deformable body includes a polymeric layer ofthickness substantially less than lateral dimensions of features beingembossed.
 33. The system of claim 25 wherein the processor determinesthe contact pressure distribution based on elastic deflections of stampand substrate.
 34. The system of claim 25 wherein the processordetermines the contact pressure distribution based on stamp indentationand substrate indentation.
 35. The system of claim 25 wherein theprocessor determines the contact pressure history distribution based ondeformations of a layer of the deformable body.
 36. The system of claim25 wherein the processor determines the contact pressure distributionbased on a zero-mean pressure distribution needed to bring all surfacesof the layer of the deformable body into contact with the stamp.
 37. Thesystem of claim 25 wherein the processor determines the contact pressuredistribution based on a zero-mean pressure distribution needed for anincremental displacement of the stamp into the deformable body.
 38. Thesystem of claim 25 wherein the processor determines the contact pressuredistribution based on a spatial variation of thickness of the layer ofthe deformable body.
 39. The system of claim 25 wherein the processordetermines the contact pressure distribution based on the point loadresponse and an average pressure applied to the stamp.
 40. The system ofclaim 25 wherein the processor determines the mechanical response of thesurface topography of the deformable body based on an average layerthickness of layers of the deformable body.
 41. The system of claim 25wherein the mechanical response of the surface topography of thedeformable body is anisotropic.
 42. The system of claim 25 wherein themechanical response of the surface topography is a spatial response. 43.The system of claim 42 wherein the mechanical response of the surfacetopography is a temporal response.
 44. A system for embossing with astamp a deformable body comprising a deformable layer on a substrate,the system comprising: a processor that runs a deformation model basedon convolving a contact pressure distribution and a point load response,the point load response being an anisotropic function; and an embosserthat embosses the deformable body using the stamp as a function of thedeformation model.
 45. The system of claim 44 wherein the deformablebody includes a polymeric layer thicker than dimensions of featuresbeing embossed.
 46. The system of claim 45 wherein the processordetermines a displacement of material in the deformable body embossedwith the stamp as a function of the point-load-time response, one ormore additional properties of the deformable body, and the contactpressure distribution.
 47. The system of claim 44 wherein the one ormore additional properties of the deformable body includes at least oneof time dependent properties, temperature dependent properties,temperature and time dependent properties, temperature dependentelasticity, or temperature dependent viscosity.
 48. The system of claim44 wherein the processor determines the anisotropic point load responseusing an anisotropy parameter that represents strength of anisotropy ofthe anisotropic layer of deformable body.